Systems and methods for determining the uncertainty in parameters of an energy use model

ABSTRACT

Systems and methods for determining the uncertainty in parameters of a building energy use model are provided. A disclosed method includes receiving an energy use model for a building site. The energy use model includes one or more predictor variables and one or more model parameters. The method further includes calculating a gradient of an output of the energy use model with respect to the model parameters, determining a covariance matrix using the calculated gradient, and using the covariance matrix to identify an uncertainty of the model parameters. The uncertainty of the model parameters may correspond to entries in the covariance matrix.

BACKGROUND

The present disclosure relates generally to systems and methods foranalyzing energy consumption model data, and more particularly tosystems and methods for determining an uncertainty in the parameters ofa building energy use model.

Many buildings are equipped with a variety of energy-consuming equipmentand devices. For example, a building may be equipped with heating,ventilation, and air conditioning (HVAC) equipment that consume energyto regulate the temperature, humidity, and/or air quality in thebuilding. Other exemplary types of energy-consuming building equipmentinclude lighting fixtures, security equipment, data networkinginfrastructure, and other such equipment.

The energy efficiency of buildings has become an area of interest inrecent years. For an energy provider, a high energy efficiency of thebuildings that it services helps to alleviate strains placed on theenergy provider's electrical generation and transmission assets. For abuilding operator, a high energy efficiency corresponds to greaterfinancial savings because less energy is consumed by the building.

One way to improve the energy efficiency of a building is through anaccurate model of the building's energy use. An energy use model for abuilding typically predicts the building's total energy consumption as afunction of one or more predictor variables and one or more modelparameters. Some energy use models include a balance point. A balancepoint (or a break even temperature) is a threshold temperature valuedefining a range of outside air temperatures for which heating orcooling is required to maintain the temperature inside the buildingwithin an acceptable range.

Previous modeling techniques allow for some of the parameters in abuilding energy use model to be estimated and for their uncertainties tobe determined. However, previous modeling techniques do not allow theuncertainty in a balance point estimate to be calculated. It remainsdifficult and challenging to determine the uncertainty associated withmany parameters in a building energy use model, including balance pointparameters.

SUMMARY

One implementation of the present disclosure is a method for determiningthe uncertainty in parameters of a building energy use model. The methodincludes receiving an energy use model for a building site. The energyuse model includes one or more predictor variables and one or more modelparameters. The method further includes calculating a gradient of anoutput of the energy use model with respect to the model parameters,determining a covariance matrix using the calculated gradient, and usingthe covariance matrix to identify an uncertainty of the modelparameters. The uncertainty of the model parameters may correspond toentries in the covariance matrix.

In some embodiments, at least one of the model parameters is a balancepoint parameter defining a range of outside air temperatures withinwhich the output of the energy use model depends on the outside airtemperature.

In some embodiments, at least one of the predictor variables is afunction of a balance point parameter of the energy use model. In suchembodiments, determining the covariance matrix may include identifying apredictor variable in the energy use model that is a function of thebalance point parameter, identifying a regression coefficient associatedwith the identified predictor variable, and using a function of theidentified regression coefficient as a gradient of the output of theenergy use model with respect to the balance point parameter. In someembodiments, the function of the identified regression coefficient is aproduct of the identified regression coefficient and a variablerepresenting a total time duration during which the outside airtemperature is within the outside air temperature range.

In some embodiments, the energy use model is at least one of anon-linear energy use model, a piecewise linear energy use model, athree-parameter energy use model, and a five-parameter energy use model.In some embodiments, determining the covariance matrix includesdetermining a standard error of regression for the energy use model.

In some embodiments, the method further includes obtaining a pluralityof data points and estimating the model parameters using the pluralityof data points. Each of the data points may include a value for the oneor more predictor variables and an associated energy consumption valuefor the building site. In some embodiments, obtaining the plurality ofdata points includes, for each of the data points, receiving at leastone of an observed temperature value and an observed enthalpy value, andcalculating the value of the predictor variable using the observedtemperature value or the observed enthalpy value.

In some embodiments, the one or more predictor variables include atleast one weather-related predictor variable. The weather-relatedpredictor variable may be at least one of cooling degree days, heatingdegree days, cooling energy days, heating energy days, temperature, andenthalpy.

In some embodiments, the method further includes updating the energy usemodel using the uncertainty in the model parameters, applying inputs tothe updated energy use model, conducting a performance analysis usingthe updated energy use model, and providing an output using a result ofthe performance analysis.

In some embodiments, the method further includes using the uncertaintyof the model parameters to perform a multivariate uncertainty analysisof the model parameters. In some embodiments, multivariate uncertaintyanalysis allows for a visualization of a correlation between two or moreof the model parameters. In some embodiments, the multivariateuncertainty analysis allows for performing a multivariate peer analysisof the model parameters.

Another implementation of the present disclosure is a method fordetermining the uncertainty in parameters of a building energy usemodel. The method includes receiving an energy use model for a buildingsite. The energy use model includes one or more predictor variables andone or more model parameters. The method further includes obtaining aset of data points and generating multiple samples from the set of datapoints. Each of the data points may include a value for the one or morepredictor variables and an associated energy consumption value for thebuilding site. Each of the samples may include a plurality of datapoints selected from the set of data points. The method furtherincludes, for each of the multiple samples, estimating the modelparameters using the plurality of data points associated with the sampleand determining an uncertainty in the model parameters using themultiple estimates of the model parameters.

In some embodiments, obtaining the set of data points includes, for eachof the data points, receiving at least one of an observed temperaturevalue and an observed enthalpy value, and calculating a value of one ormore of the predictor variables using the observed temperature value orthe observed enthalpy value.

In some embodiments, the one or more predictor variables include atleast one weather-related predictor variable. The weather-relatedpredictor variable may be at least one of cooling degree days, heatingdegree days, cooling energy days, heating energy days, temperature, andenthalpy.

In some embodiments, at least one of the model parameters is a balancepoint parameter defining a range of outside air temperatures withinwhich an output of the energy use model depends on the outside airtemperature.

In some embodiments, generating multiple samples from the set of datapoints includes, for each of the multiple samples, replacing apredetermined number of data points in the set of data points with anequal number of data points selected randomly from the set of datapoints. Each of the multiple samples may include the same number of datapoints as the original set of multiple data points.

In some embodiments, generating multiple samples from the set of datapoints includes, for each of the multiple samples, removing apredetermined number of data points from the set of multiple datapoints. Each of the multiple samples may include less data points thanthe set of data points. In some embodiments, the predetermined number ofdata points is one data point and each of the multiple samples includesone less data point than the original set of multiple data points.

In some embodiments, the method further includes updating the energy usemodel using the uncertainty in the model parameters, applying inputs tothe updated energy use model, conducting a performance analysis usingthe updated energy use model, and providing an output using a result ofthe performance analysis.

Another implementation of the present disclosure is a system fordetermining the uncertainty in parameters of a building energy usemodel. The system includes a data storage device and a processingcircuit configured to receive an energy use model for a building siteand to store the energy use model in the data storage device. The energyuse model may include one or more predictor variables and one or moremodel parameters. The processing circuit is further configured tocalculate a gradient of an output of the energy use model with respectto the model parameters, to determine a covariance matrix using thecalculated gradient, and to use the covariance matrix to identify anuncertainty of the model parameters. The uncertainty of the modelparameters may correspond to entries in the covariance matrix.

In some embodiments, the energy use model is at least one of anon-linear energy use model, a piecewise linear energy use model, athree-parameter energy use model, and a five-parameter energy use model.

In some embodiments, at least one of the model parameters is a balancepoint parameter defining a range of outside air temperatures withinwhich the output of the energy use model depends on the outside airtemperature.

In some embodiments, at least one of the predictor variables is afunction of a balance point parameter of the energy use model. In suchembodiments, determining the covariance matrix may include identifying,in the energy use model, a predictor variable that is a function of thebalance point parameter, identifying a regression coefficient associatedwith the identified predictor variable, and using a function of theidentified regression coefficient as a gradient of the output of theenergy use model with respect to the balance point parameter. In someembodiments, the function of the identified regression coefficient is aproduct of the identified regression coefficient and a variablerepresenting a total time duration during which the outside airtemperature is within the outside air temperature range.

In some embodiments, the processing circuit is further configured toupdate the energy use model using the uncertainty in the modelparameters, apply inputs to the updated energy use model, conduct aperformance analysis using the updated energy use model, and provide anoutput using a result of the performance analysis.

In some embodiments, the processing circuit is further configured to usethe uncertainty of the model parameters to perform a multivariateuncertainty analysis of the model parameters. In some embodiments,multivariate uncertainty analysis allows for a visualization of acorrelation between two or more of the model parameters. In someembodiments, the multivariate uncertainty analysis allows for performinga multivariate peer analysis of the model parameters.

Another implementation of the present disclosure is a method foranalyzing an energy performance of a building site. The method includesdetermining an uncertainty in a break even temperature parameter of anenergy use model for the building site and using the uncertainty in thebreak even temperature parameter to analyze the energy performance ofthe building site.

In some embodiments, determining the uncertainty in the break eventemperature parameter includes simultaneously determining theuncertainty of the break even temperature parameter and an uncertaintyof one or more additional parameters of the energy use model and usingthe simultaneously determined uncertainties to improve an accuracy ofthe energy performance analysis.

In some embodiments, analyzing the energy performance of the buildingsite includes performing a peer analysis of one or more energy use modelparameters for a class of buildings including the building site. Thepeer analysis may include calculating a difference between an energy usemodel parameter for the building site and a mean of the energy use modelparameters for the class of buildings, detecting an outlier modelparameter based on a result of the calculation, and using theuncertainty in the break even temperature parameter to improve anaccuracy of the detection.

In some embodiments, analyzing the energy performance of the buildingsite includes performing a temporal analysis of one or more energy usemodel parameters for the building site. The temporal analysis mayinclude calculating a difference between a value for an energy use modelparameter at a particular time and a mean of a set of values for theenergy use model parameter. The set of values may include a plurality ofvalues for the energy use model parameter at various times. The temporalanalysis may further include using the uncertainty in the break eventemperature parameter to improve an accuracy of the calculation anddetecting an outlier model parameter based on a result of thecalculation.

In some embodiments, analyzing the energy performance of the buildingsite includes monitoring changes to one or more parameters in the energyuse model for the building site, detecting the existence of a faultcondition using a monitored change to the energy use model parameters,and using the uncertainty in the break even temperature parameter toimprove an accuracy of the detection.

In some embodiments, analyzing the energy performance of the buildingsite includes calculating an energy savings opportunity for the buildingsite and using the uncertainty in the break even temperature parameterto improve an accuracy of the calculation.

In some embodiments, the method further includes updating the energy usemodel using the uncertainty in the model parameters, applying inputs tothe updated energy use model, conducting a performance analysis usingthe updated energy use model, and providing an output using a result ofthe performance analysis.

Another implementation of the present disclosure is one or morenon-transitory computer-readable media having computer-executableinstructions stored therein. The instructions, when executed by one ormore processors may cause the one or more processors to performoperations including receiving an energy use model for a building site.The energy use model includes one or more predictor variables and one ormore model parameters. The operations further include calculating agradient of an output of the energy use model with respect to the modelparameters, determining a covariance matrix using the calculatedgradient, and using the covariance matrix to identify an uncertainty ofthe model parameters. The uncertainty of the model parameters maycorrespond to entries in the covariance matrix.

Another implementation of the present disclosure is one or morenon-transitory computer-readable media having computer-executableinstructions stored therein. The instructions, when executed by one ormore processors may cause the one or more processors to performoperations including receiving an energy use model for a building site.The energy use model includes one or more predictor variables and one ormore model parameters. The operations further include obtaining a set ofdata points and generating multiple samples from the set of data points.Each of the data points may include a value for the one or morepredictor variables and an associated energy consumption value for thebuilding site. Each of the samples may include a plurality of datapoints selected from the set of data points. The operations furtherinclude, for each of the multiple samples, estimating the modelparameters using the plurality of data points associated with the sampleand determining an uncertainty in the model parameters using themultiple estimates of the model parameters.

Another implementation of the present disclosure is one or morenon-transitory computer-readable media having computer-executableinstructions stored therein. The instructions, when executed by one ormore processors may cause the one or more processors to performoperations including determining an uncertainty in a break eventemperature parameter of an energy use model for the building site andusing the uncertainty in the break even temperature parameter to analyzethe energy performance of the building site. In some embodiments,determining the uncertainty in the break even temperature parameterincludes simultaneously determining the uncertainty of the break eventemperature parameter and an uncertainty of one or more additionalparameters of the energy use model and using the simultaneouslydetermined uncertainties to improve an accuracy of the energyperformance analysis.

Those skilled in the art will appreciate that the summary isillustrative only and is not intended to be in any way limiting. Otheraspects, inventive features, and advantages of the devices and/orprocesses described herein, as defined solely by the claims, will becomeapparent in the detailed description set forth herein and taken inconjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing of a building data acquisition system including abuilding analysis system configured to collect, store, and/or analyzeperformance data related to a building's energy use, shown according toan exemplary embodiment.

FIG. 2 is a drawing of an energy use profile for a building siteillustrating a heating balance point, a heating slope, a cooling balancepoint, and a cooling slope, according to an exemplary embodiment.

FIG. 3 is a block diagram illustrating the building analysis system ofFIG. 1 in greater detail, according to an exemplary embodiment.

FIG. 4 is a flowchart of a process for determining the values anduncertainties of parameters of an energy use model using an analyticaltechnique, according to an exemplary embodiment.

FIG. 5 is a flowchart of a process for determining the values anduncertainties of parameters of an energy use model using an empiricaltechnique, according to an exemplary embodiment.

FIG. 6 is a flowchart of a process for analyzing the energy performanceof a building site using an estimate of the uncertainty associated withone or more balance point parameters, according to an exemplaryembodiment.

DETAILED DESCRIPTION

Referring generally to the FIGURES, systems and methods for determiningthe uncertainty in parameters of a building energy use model andcomponents thereof are shown, according to various exemplaryembodiments. A building energy use model generally describes the energyusage of a building or building site in terms of one or more predictorvariables (e.g., weather data, occupancy data, etc.) and one or moremodel parameters. Building energy use models may take a variety of formsincluding parametric models (e.g., linear regression, non-linearregression, etc.), non-parametric models (e.g., neural networks, kernelestimation, Bayesian, etc.) or something in between (e.g., Gaussianprocess models, etc.).

Some building energy use models include a balance point. A balance point(or a break even temperature) is a threshold temperature value defininga range of outside air temperatures for which heating or cooling isrequired to maintain the temperature inside the building within anacceptable range. For example, a cooling balance point may be athreshold outside air temperature above which cooling is required and aheating balance point may be a threshold outside air temperature belowwhich heating is required. The value of the balance point may be used tocalculate the value of a weather-related predictor variable (e.g.,cooling degree days (CDD), heating degree days (HDD), etc.) used in achange-point energy use model.

The systems and methods described herein may be used to determine theuncertainty in the balance points and/or other parameters of a buildingenergy use model. In some embodiments, a regression technique is used toestimate values for coefficients of the energy use model (i.e., theregression model coefficients) and one or more balance points (e.g., aheating balance point and/or a cooling balance point). The balancepoints may be added to a parameter vector containing the regressionmodel coefficients. Unlike traditional techniques which estimateuncertainty values for only the regression model coefficients, thesystems and methods of the present disclosure allow for estimating theuncertainties in the values of both the regression model coefficientsand the balance points simultaneously. As used herein, the term “modelparameters” describes all of the parameters of a building energy usemodel including both regression model coefficients and balance pointparameters.

Adding the balance points to the parameter vector containing theregression model coefficients allows the uncertainties in the balancepoints to be determined along with the uncertainties of the regressionmodel coefficients. For example, a gradient of an output of the energyuse model (e.g., a predicted energy consumption) with respect to themodel parameters can be calculated and used to generate a covariancematrix. The uncertainty in the balance point parameters may correspondto entries in the covariance matrix and may be identified by inspectionthereof.

Estimating the uncertainties in the regression model coefficients andthe balance points simultaneously results in a more accurate calculationof the uncertainties in the model parameters compared with theuncertainties obtained using traditional regression techniques.Estimating the uncertainties in the regression model coefficients andthe balance points simultaneously also allows for a multivariateanalysis. Using the systems and methods of the present disclosure, theuncertainties in the balance point parameters can be calculated and theuncertainties in the regression model coefficients can be determinedmore accurately than with traditional techniques.

Referring now to FIG. 1, an illustration of a building data acquisitionsystem 100 is shown, according to an exemplary embodiment. Building dataacquisition system 100 may be configured to collect, store, and/oranalyze performance data related to a building's energy use. Theperformance data for the building may be used, for example, to model thebuilding's energy usage, predict related parameters in the energy usemodel, determine an appropriate parameter order for the energy usemodel, and/or estimate the uncertainties associated with various modelparameters.

Building data acquisition system 100 is shown to include buildings102-106. Buildings 102-106 may include any number of buildings (e.g., afirst through a nth building) and any type of buildings (e.g.,commercial buildings, residential buildings, industrial buildings,etc.). For example, building 102 may be an office building, building 104may be a manufacturing facility, and building 106 may be a hospitalityfacility, such as a hotel. Other exemplary buildings in buildings102-106 may include, but are not limited to, data centers, schools,shipping facilities, and government buildings. Buildings 102-106 mayinclude any combination of building types.

Buildings 102-106 may be located within the same geographic regions asone another or across different geographic regions. For example,building 102 and building 104 may be located in the same city, whilebuilding 106 may be located in a different city. Different levels ofgranularity may be used to distinguish buildings 102-106 as beinglocated in the same geographic region. For example, geographic regionsmay be divided by country, state, city, metropolitan area, time zone,zip code, area code, latitude, longitude, growing zone, combinationsthereof, or using any other geographic classification system. In someembodiments, a building's geographic location may be used as a proxy forits climatic zone. For example, data regarding a building's location inHawaii may be used to determine that the building is located in atropical climate.

In some embodiments, each of buildings 102-106 may be part of a buildingsite. A building site may include one or more of buildings 102-106 andtypically includes buildings that are located proximate to each otherand/or interconnected. A single HVAC system, water system, and/orelectric grid may service multiple buildings that are part of the samebuilding site. An energy use model may be developed for individualbuildings, for a building site, or for both individual buildings and abuilding site.

Buildings 102-106 may be equipped with sensors and other monitoringdevices configured to measure performance data related to the building'senergy consumption. For example, buildings 102-106 may have devices(e.g., computing devices, power meters, etc.) configured to measure thewater consumption, energy consumption, and energy demand of buildings102-106. Other forms of performance data may include a measuredtemperature in one or more zones of a building, dimensions of thebuilding (e.g., square footage, etc.), and/or any other value thatrelates to the building's energy usage profile. In some embodiments,performance data includes data used in a building automation system. Forexample, performance data may include control parameters (e.g., setpoints, tuning parameters, threshold values, etc.) used to regulate thetemperature in the building and/or timing data used to automaticallyturn on or off lighting within the building (e.g., at night, when thebuilding is unoccupied, according to a set schedule, etc.).

In some embodiments, readily available data may be used to determine andmodel a building's energy consumption. For example, billing datareceived from a utility 114 may be used to determine a building's energyconsumption and the financial costs associated with the energyconsumption. Such an approach may simplify and reduce the cost ofperforming the energy analysis over approaches that rely heavily onsensor data from a building.

In some embodiments, performance data includes weather data for a regionin which buildings 102-106 are located. The weather data may begenerated by weather-sensing equipment at buildings 102-106. Forexample, buildings 102-106 may be equipped with temperature sensors thatmeasure the outside air temperature. In other embodiments, buildings102-106 may be configured to receive weather data from an externalweather data source.

In some embodiments, performance data includes weather data for atypical meteorological year (TMY) received from historical weather datasource 112 (e.g., a computer system of the National Oceanic andAtmospheric Administration or similar data source). In the United Statesof America, the first set of TMY data was collected between 1948-1980from various locations throughout the country. A second set of TMY data(TMY2), which also includes data regarding precipitable moisture, wascollected between 1961-1990. In addition, a third set of TMY data(TMY3), was collected from many more locations than TMY2 data over thespan of 1976-1995. Regardless of the version used, TMY data may be usedto compare current conditions to normal or predicted conditions, in someembodiments. In further embodiments, TMY data may be used to predictfuture conditions of a building (e.g., by using the historical data topredict typical future weather conditions) or future energy consumptionsby a building. For example, TMY data may be used to predict an averageoutdoor temperature change for a building during the upcoming month ofMarch. TMY data may be stored by the building automation systems ofbuildings 102-106 or building analysis system 110 and used to model theheating and cooling needs of buildings 102-106. As used herein, “TMYdata” may refer to any version or set of TMY data (e.g., TMY2 data, TMY3data, etc.).

Performance data may be collected for individual buildings 102-106 orfor a building site. For example, energy usage data (e.g., received fromutility 114 or otherwise) for multiple buildings may be combined into atotal energy usage for a building site. Buildings that are part of thesame building site may share the same outside air temperature, outsideair enthalpy, or other weather-related predictor variables (e.g.,cooling degree days, heating degree days, cooling energy days, heatingenergy days, etc.).

Still referring to FIG. 1, building data acquisition system 100 is shownto include a network 108. Network 108 may be any form of computernetwork that relays information between buildings 102-106 and a buildinganalysis system 110. For example, network 108 may include the Internetand/or other types of data networks, such as a local area network (LAN),a wide area network (WAN), a cellular network, satellite network, orother types of data networks. Network 108 may also include any number ofcomputing devices (e.g., computer, servers, routers, network switches,etc.) that are configured to receive and/or transmit data within network108. Network 108 may further include any number of hardwired and/orwireless connections. For example, building 102 may communicatewirelessly (e.g., via WiFi, ZigBee, cellular, radio, etc.) with atransceiver that is hardwired (e.g., via a fiber optic cable, a CAT5cable, etc.) to other computing devices in network 108.

Still referring to FIG. 1, building data acquisition system 100 is shownto include a building analysis system 110. Building analysis system 110may include one or more electronic devices connected to network 108. Invarious embodiments, building analysis system 110 may be a computerserver (e.g., an FTP server, file sharing server, web server, etc.) or acombination of servers (e.g., a data center, a cloud computing platform,etc.). Building analysis system 110 may include a processing circuitconfigured to perform the functions and processes described herein.

Building analysis system 110 may be configured to obtain performancedata for buildings 102-106 (e.g., either directly from buildings 102-106or from another computing device connected to network 108). Theperformance data may include a plurality of data points. Each of thedata points may include a value for a predictor variable (e.g., outsideair temperature or enthalpy, cooling degree days, heating degree days,cooling energy days, heating energy days, etc.) and an associated energyconsumption value for a building or building site. The performance datamay be received by building analysis system 110 periodically, inresponse to a request for data from building analysis system 110, inresponse to receiving a request from a client device 116 (e.g., a useroperating client device 116 may request that the building data be sentby the computing device), or at any other time.

In some embodiments, building analysis system 110 is configured to modelthe energy usage of buildings 102-106 using the performance data. Inother embodiments, building analysis system receives an energy use modelfor buildings 102-106 from an external source (e.g., within buildingdata acquisition system 100 or otherwise). The building energy use modelgenerated or received by building analysis system 110 may be aparametric model or a non-parametric model. In some embodiments,building analysis system 110 may perform LEAN energy analysis usingreadily available data (e.g., utility billing data, weather data, etc.)to model the energy usage profiles of buildings 102-106 and/or topredict an energy cost for buildings 102-106. Building analysis system110 may generate and provide various reports to client 116, which may belocated within one of buildings 102-106 or at another location.

In some embodiments, building analysis system 110 may be implemented atone or more of buildings 102-106. For example, building analysis system110 may be integrated as part of a building automation system (BAS) forbuildings 102-106 (e.g., as part of a centralized BAS or in adistributed implementation). In a distributed implementation,performance data may be shared among the distributed components ofbuilding analysis system 110 via network 108. For example, computingdevices at buildings 102-106 may be configured to collaboratively shareperformance data regarding their respective building's energyconsumption and demand. The sharing of performance data among thebuildings' respective computing devices may be coordinated by one ormore of the devices, or by a remote coordination service (e.g., asupervisory controller or remote server connected to network 108).Building analysis system 110 is described in greater detail withreference to FIG. 3.

Referring now to FIG. 2, an energy use profile 200 for a building orbuilding site is shown, according to an exemplary embodiment. Ingeneral, a number of different factors may affect the energy use of abuilding. For example, weather-related variables such as the airtemperature outside the building may affect the amount of energyrequired to heat or cool the building to a setpoint temperature. In someembodiments, the building's energy use profile when cooling the buildingmay differ from the building's energy use profile when heating thebuilding. In some embodiments, the energy use model for the buildingincludes parameters relating to both heating and cooling the building(e.g., for a four-parameter model or a five-parameter model). In otherembodiments, the energy use model for the building includes parametersrelating to either heating or cooling the building, but not both (e.g.,for a two-parameter model or a three parameter model).

Energy use profile 200 is shown as an x-y plot with building energy useE plotted along a first axis 202 and outdoor air temperature T_(OA)plotted along a second axis 204. In various embodiments, the building'senergy use E may be an energy consumption (e.g., measured in kWh) or anenergy cost associated with the building's energy consumption (e.g.,measured in dollars or other units of currency). Energy consumptionand/or energy cost information may be obtained, for example, frombilling data provided by utility 114. In some embodiments, the outdoorair temperature T_(OA) may be measured using sensors located at or nearthe building over a particular time period. Energy use profile 200 isshown to include a base energy load E₀ 206, a heating balance pointT_(bH) 208, a cooling balance point T_(bC) 210, a heating slope S_(H)212, and a cooling slope S_(C) 214.

Base energy load E₀ 206 may be a baseline or fixed energy usage thatdoes not depend on the outdoor air temperature T_(OA). For example, baseenergy load E₀ 206 may be a function of the energy consumption of thebuilding's lighting, computer systems, security systems, and other suchelectronic devices in the building. Since the energy consumption ofthese devices does not change as a function of the outdoor airtemperature T_(OA), base energy load E₀ 206 may be used to represent theportion of the building's energy consumption that is not a function ofthe outdoor air temperature T_(OA).

Heating slope S_(H), 212 may correspond to the change in energyconsumption or energy costs that results when the outdoor airtemperature T_(OA) drops below a heating balance point T_(bH) 208 (e.g.,a break even temperature). For example, assume that heating balancepoint T_(bH) 208 for a building is 55° F. When the outdoor airtemperature T_(OA) is at or above 55° F., only an energy expenditureequal to base load E₀ 206 may be needed to maintain the internaltemperature of the building within an acceptable temperature range.However, additional energy may be needed if the outdoor air temperatureT_(OA) drops below 55° F. (e.g., to provide mechanical heating to theinterior of the building). As the outdoor air temperature T_(OA)decreases, the amount of energy needed to heat the building increases ata rate corresponding to heating slope S_(H) 212.

Cooling slope S_(C) 214 may correspond to the change in energyconsumption or energy costs that result when the outdoor air temperatureT_(OA) rises above a cooling balance point T_(bC) 210 (e.g., a breakeven temperature). For example, assume that cooling balance point T_(bC)210 for a building is 67° F. When the outdoor air temperature T_(OA) isat or below 67° F., only an energy expenditure equal to base load E₀ 206may be needed to maintain the internal temperature of the building.However, additional energy may be needed if the outdoor air temperatureT_(OA) rises above 67° F. (e.g., to provide mechanical cooling to theinterior of the building). As the outdoor air temperature T_(OA)increases, the amount of energy needed to cool the building increases ata rate corresponding to cooling slope S_(C) 214.

Still referring to FIG. 2, energy use profile 200 may be associated witha building or building site having a five-parameter energy use model,where base energy usage E₀ 206, heating balance point T_(bH) 208,cooling balance point T_(bC) 210, heating slope S_(H) 212, and coolingslope S_(C) 214 correspond to the five parameters of the five-parametermodel.

In some embodiments, not all five parameters may be necessary orappropriate to model a building's energy use. For example, if thebuilding is located in a cold climate such that the outdoor airtemperature T_(OA) is never higher than heating balance point T_(bH)208, a two-parameter heating model may be appropriate. Parameters in thetwo-parameter heating model may include base energy load E₀ 206 andheating slope S_(H) 212. An energy profile associated with atwo-parameter heating model may be the portion of energy use profile 200to the left of two-parameter heating line 216.

If the building is located in a hot climate such that the outdoor airtemperature T_(OA) is never lower than cooling balance point T_(bC) 210,a two-parameter cooling model may be appropriate. Parameters in thetwo-parameter cooling model may include base energy load E₀ 206 andcooling slope S_(C) 214. An energy profile associated with atwo-parameter cooling model may be the portion of energy use profile 200to the right of two-parameter cooling line 220.

If the building is located in a moderately cool climate such that theoutdoor air temperature T_(OA) is sometimes below heating balance pointT_(bH) 208 (e.g., T_(OA)<T_(bH)) and sometimes between heating balancepoint T_(bH) 208 and cooling balance point T_(bC) 210 (e.g.,T_(bH)<T_(OA)<T_(bC)), or there is not a cooling balance point orcooling does not use the energy type being modeled, a three-parameterheating model may be appropriate. Parameters in the three-parameterheating model may include base energy load E₀ 206, heating balance pointT_(bH) 208, and heating slope S_(H) 212. An energy profile associatedwith a three-parameter heating model may be the portion of energy useprofile 200 to the left of three-parameter heating line 218.

If the building is located in a moderately warm climate such that theoutdoor air temperature T_(OA) is sometimes above cooling balance pointT_(bC) 210 (e.g., T_(OA)>T_(bC)) and sometimes between heating balancepoint T_(bH) 208 and cooling balance point T_(bC) 210 (e.g.,T_(bH)<T_(OA)<T_(bC)), or there is not a heating balance point orheating does not use the energy type being modeled, a three-parametercooling model may be appropriate. Parameters in the three-parametercooling model may include base energy load E₀ 206, cooling balance pointT_(bC) 210, and cooling slope S_(C) 214. An energy profile associatedwith a three-parameter cooling model may be the portion of energy useprofile 200 to the right of three-parameter cooling line 222.

If the building transitions between supplying heating and cooling at asingle balance point (e.g., the building's heating balance point T_(bH)and cooling balance point T_(bC) are equal), a four parameter model maybe appropriate. The parameters in the four-parameter model may includebase energy load E₀ 206, heating slope S_(H) 212, cooling slope S_(C)214, and a single balance point which is both heating balance pointT_(bH) 208 and cooling balance point T_(bC) 210.

Referring now to FIG. 3, a block diagram illustrating a buildinganalysis system 110 in greater detail is shown, according to anexemplary embodiment. Building analysis system 110 may be configured toreceive an energy use model for a building site. The energy use modelmay have one or more predictor variables and one or more modelparameters. Building analysis system 110 may calculate a gradient of anoutput of the energy use model (e.g., predicted energy consumption) withrespect to the model parameters and determine a covariance matrix usingthe calculated gradient. Building analysis system 110 may use thecovariance matrix to identify an uncertainty of the model parameters(e.g., regression coefficients of the energy use model, one or morebalance points, etc.). For example, the uncertainties of the modelparameters may correspond to various entries in the covariance matrixand building analysis system 110 may identify the various uncertaintiesby inspection thereof.

Building analysis system 110 is shown to include a communicationsinterface 302, a user interface I/O 303, and a processing circuit 304.Communications interface 302 may include wired or wireless interfaces(e.g., jacks, antennas, transmitters, receivers, transceivers, wireterminals, etc.) for conducting electronic data communications with thevarious components of building data acquisition system 100 or otherexternal devices or data sources. Data communications may be conductedvia a direct connection (e.g., a wired connection, an ad-hoc wirelessconnection, etc.) or a network connection (e.g., an Internet connection,a LAN, WAN, or WLAN connection, etc.). For example, communicationsinterface 302 can include an Ethernet card and port for sending andreceiving data via an Ethernet-based communications link or network. Invarious embodiments, communications interface 302 can include a WiFitransceiver, a cellular transceiver, or a mobile phone transceiver forcommunicating via a wireless communications network.

Communications interface 302 may receive energy-related performance datafor a building or building site. Performance data may include, forexample, energy consumption data, energy cost data, energy demand data,outside air temperature data, historical weather or meteorological data,pricing or billing data (e.g., from an energy provider), predictedenergy usage data, or other energy-related data associated with abuilding or building site. In some embodiments, the performance dataincludes a plurality of data points including at least oneweather-related predictor variable (e.g., outside airtemperature/enthalpy, cooling degree days, heating degree days, coolingenergy days, heating energy days, etc.).

In some embodiments, communications interface 302 receives an energy usemodel for the building or building site. The energy use model may haveknown model parameters or unknown model parameters. In otherembodiments, building analysis system 110 constructs the energy usemodel using the energy-related performance data. In some embodiments,communications interface 302 receives an indication of an appropriatemodel parameter order and/or an indication of one or more specificparameters to include in the energy use model. Building analysis system110 may determine values for the model parameters (e.g., using aregression technique or any other system identification process).

Still referring to FIG. 3, building analysis system 110 is shown toinclude a user interface I/O 303. User interface I/O 303 may include oneor more user interface input and/or output devices for facilitating userinteraction with building analysis system 110. User interface I/O 303may include, for example, a local display (e.g., a LCD panel, anelectronic display screen, one or more indicator lights, etc.), akeyboard, a mouse, a printer, a microphone, a speaker, a touch-sensitivepanel, a camera, a scanner, one or more user-operable buttons, dials,sliders, switches, or any other type of user interface device.

User interface I/O 303 may be used to receive input from a user (e.g.,physical input, verbal input, etc.) and to provide output to a user in auser-comprehensible format (e.g., text, numbers, words, sounds, statusindicators, visual displays, printouts, etc.). For example, a user mayinteract with user interface I/O 303 to submit a request for informationregarding the parameter order of a particular building energy use model.Building analysis system 110 may process the user request and providethe user with an output (e.g., a visual display, a textual/graphicaloutput, etc.) indicating the parameter order of the particular buildingenergy use model. As another example, a user may interact with userinterface I/O to request a performance analysis report for a particularbuilding or building system. Building analysis system 110 may processthe request and provide the user with an output (e.g., a visual display,a textual/graphical report, etc.) analyzing the performance of theparticular building or building system.

In various embodiments, user input may be received locally (e.g., viauser interface I/O 303) or remotely (e.g., via a LAN connection, a WANconnection, a network connection, an Internet connection, etc.) from aremote user interface client (e.g., a remote computer, a remote userdevice, etc.). User output may also be provided locally to a userinteracting with building analysis system 110 via user interface I/O 303or remotely to a user interacting with building analysis system 110 viaa remote user interface client (e.g., a remote computer, over a network,etc.). In some embodiments, user input and user output may be sent andreceived via communications interface 302, user interface I/O 303,and/or a combination of both communications interface 302 and userinterface I/O 303.

Still referring to FIG. 3, building analysis system 110 is shown toinclude a processing circuit 304. In some embodiments, processingcircuit 304 is a component of building analysis system 110. In otherembodiments, processing circuit 304 is a component of any othercomputing device or system configured to analyze energy-relatedcharacteristics and/or statistics of a building site. In someembodiments, the various components of processing circuit 304 may bedistributed across multiple computing devices or systems.

Processing circuit 304 is shown to include a processor 306 and memory308. Processor 306 can be implemented as one or more microprocessors(e.g., CPUs, GPUs, etc.), an application specific integrated circuit(ASIC), one or more field programmable gate arrays (FPGAs), a circuitcontaining one or more processing components, a group of distributedprocessing components (e.g., processing components in communication viaa data network or bus), circuitry for supporting a microprocessor, orother hardware configured for processing data. Processor 306 may beconfigured to execute computer code stored in memory 308 to complete andfacilitate the activities described herein.

Memory 308 may include one or more devices (e.g., RAM, ROM, solid statememory, hard disk storage, etc.) for storing data and/or computer codefor completing or facilitating the various processes, layers, andmodules of the present disclosure. Memory 308 may include volatilememory or non-volatile memory. Memory 308 may include databasecomponents, object code components, script components, or any other typeof information structure for supporting the various activities andinformation structures of the present disclosure. According to anexemplary embodiment, memory 308 is communicably connected to processor306 via processing circuit 304 and includes computer code for executing(e.g., by processing circuit 304 and/or processor 306) one or moreprocesses described herein. In brief overview, memory 308 is shown toinclude a building data module 310, a predictor variable module 312, anenergy use model module 314, a vector augmentation module 316, an outputgradient module 318, a covariance matrix module 320, an uncertaintyidentification module 322, a data sampling module 324, an empiricalanalysis module 326, an energy analysis module 328, an output and clientrequest module 330, and a building control services module 332.

Still referring to FIG. 3, memory 308 is shown to include a buildingdata module 310. Building data module 310 may obtain and/or storebuilding data related to buildings 102-106. In some embodiments,building data includes data relating to the physical characteristics ofa building. For example, building data may include data regarding abuilding's geographic location (e.g., street address, city, coordinates,etc.), dimensions (e.g., floor space, stories, etc.), classification(e.g., office space, hospital, school, etc.), building materials, or anyother physical characteristic which may be used to describe a building.

In some embodiments, building data includes energy-related performancedata for buildings 102-106. Energy-related performance data may include,for example energy consumption data (e.g., current energy usage,historical energy usage, predicted energy usage, etc.), measuredtemperatures or other sensory data obtained by one or more sensorydevices of buildings 102-106, and/or control parameters (e.g., setpoints, tuning parameters, threshold values, etc.) used to regulate thetemperature or other variables within buildings 102-106. In someembodiments, building data includes baseline energy consumption data(e.g., a base load E₀), balance point data (e.g., a heating balancepoint T_(bH), a cooling balance point T_(bC), a single balance pointwhich is both the heating balance point T_(bH) and the cooling balancepoint T_(bC), etc.), heating or cooling slope data (e.g., a heatingslope S_(H), a cooling slope S_(C), etc.), or other data describing theparameters used in an energy use model for a particular building orbuilding site.

In some embodiments, building data may include billing data from one ormore utilities (e.g., utility 114) that supply buildings 102-106 with aconsumable resource. For example, building data may include billing datafrom a utility that provides the building with electrical power. Inanother example, building data may include billing data from a utilitythat supplies water to the building.

In some embodiments, building data module 310 uses the building data tocalculate one or more normalized metrics. For example, building datamodule 310 may normalize the building's energy consumption using thebuilding's internal volume or area. The normalized energy consumptionmay be expressed as an energy consumption per unit area

$( {{e.g.},\frac{kWh}{{ft}^{2}}} )$

and/or an energy consumption per unit volume

$( {{e.g.},\frac{kWh}{{ft}^{3}}} ).$

The normalized metrics may be used by building analysis system 110 tocompare the energy consumption of buildings having different sizes,areas, and/or volumes.

Building data module 310 may store a plurality of energy consumptionvalues in a data set. The energy consumption values may correspond to aplurality of observations of the building's energy consumption (e.g., atvarious times, at various locations, etc.). In some embodiments,building data module 310 stores energy consumption observations in anenergy consumption vector Y. Energy consumption vector Y may have a sizeof n×1 (i.e., a single-dimensional vector), where n is the total numberof observations of the building's energy consumption or power use.

Still referring to FIG. 3, memory 308 is shown to include a predictorvariable module 312. Predictor variable module 312 may obtain and storedata relevant to one or more predictor variables used in an energy usemodel for a building or building site. In some embodiments, predictorvariable data includes weather data for one or more geographiclocations. For example, weather data may include historical, current, orpredicted data regarding a location's temperature (e.g., outside airtemperature), humidity, atmospheric pressure, wind speed, precipitablewater, or other weather-related data. In some embodiments, weather datamay be gathered via sensors located at or near buildings 102-106. Insome embodiments, weather data includes TMY data (e.g., TMY2 data, TMY3data, etc.). Weather data may include weather data from any number ofdifferent time periods having any degree of granularity. For example,weather data may identify weather conditions on a monthly, weekly,daily, or hourly level.

In some embodiments, predictor variable module 312 uses the weather datato calculate a value for a weather-related predictor variable. Theweather-related predictor variable may be any variable that depends on aweather-related value (e.g., outside air temperature T_(OA), enthalpy,humidity, pressure, wind speed, precipitation level, etc.). For example,the weather-related predictor variable may be a cooling degree day (CDD)value, a heating degree day (HDD) value, a cooling energy day (CED)value, or a heating energy day (HED) value.

A CDD or HDD value may represent the amount of heating or cooling neededby the building over a period of time. In some embodiments, predictorvariable module 312 calculates the CDD and HDD values for a building byintegrating the difference between the outside air temperature T_(OA) ofthe building and a given temperature over a period of time. The giventemperature may be a cooling balance point for the building (e.g.,cooling balance point T_(bC) 210) to determine a CDD value, or heatingbalance point for the building (e.g., heating balance point T_(bH) 208)to determine a HDD value. For example, CDD and HDD values for thebuilding over the course of a month may be calculated as follows:

CDD=∫ ^(month)Max{0,(T _(OA) −T _(bC))}dt

HDD=∫ ^(month)Max{0,(T _(bH) −T _(OA))}dt

In some embodiments, the value for one or more of the predictorvariables is a function of a heating balance point or a cooling balancepoint. In other embodiments, a set reference temperature may be used tocalculate a building's CDD or HDD value instead of the building's actualbalance point. For example, a reference temperature of 65° F. may beused as a fixed value to compare with the building's outdoor airtemperature. CED and HED values may be calculated in a similar mannerusing outside air enthalpy rather than outside air temperature T_(OA).

In some embodiments, predictor variable module 312 obtains and storesdata relating to one or more non-weather-related predictor variables.Non-weather-related predictor variables may include, for example, waterconsumption, building occupancy, days off, the number of days perperiod, and/or any other variable which may affect the building's energyconsumption.

Predictor variable module 312 may store values for each predictorvariable in a data set. In some embodiments, predictor variable module312 stores the values for the predictor variables in a predictorvariable matrix X. The predictor variable matrix X may have a size of nby p+1, where n is the total number of observations and p+1 is the totalnumber of predictor variables including a time variable t indicating theduration of each observation period. (e.g., t=[t₁ . . . t_(n)]^(T)).

Still referring to FIG. 3, memory 308 is shown to include an energy usemodel module 314. Energy use model module 314 may store one or moreenergy use models for a building or building site. The one or moreenergy use models may be of any form. For example, energy use modelmodule 314 may store parametric models (e.g., linear regression models,non-linear regression models, etc.), non-parametric models (neuralnetworks, kernel estimation, hierarchical Bayesian, etc.), or somethingin between (e.g., Gaussian process models). In some embodiments, energyuse model module 314 receives one or more energy use models viacommunications interface 302. In other embodiments, energy use modelmodule 314 generates one or more energy use models using the energyconsumption data stored in building data module 310 (e.g., energyconsumption vector Y) and the predictor variable data stored inpredictor variable module 312 (e.g., predictor variable matrix X).

In some embodiments, energy use model module 314 models the energy useof a building using linear regression. A linear regression model for abuilding may be represented by the following equation:

Y=Xβ+e,

where Y is the building's energy consumption, X is a predictor variablematrix, β is a vector of unknown regression coefficients having a sizeof p+1 (e.g., β₀, β₁, . . . β_(p)), and e is the model error such thate˜N(0, t_(k)σ²).

The predictor variable matrix X may have a size of n by p+1 where n isthe total number of observations and where the number p+1 includes ppredictor variables (e.g., x₁ . . . x_(p)) and a time variable trepresenting the duration of each observation period (e.g., t=[t₁ . . .t_(n)]^(T)). For example, the estimated energy consumption during thek^(th) observation period can be expressed as:

ŷ _(k)=β₀ t _(k)+β₁ x _(1,k)+β₂ x _(2,k)+ . . . +β_(p) x _(p,k) +e _(k),

where t_(k) is the time duration of the k^(th) observation period k=1 .. . n.

In some embodiments, modeling the energy use of a building using linearregression includes estimating the values of the parameter vector β.Energy use model module 314 may use any of a variety of differentestimation techniques to estimate the values of the parameter vector β.In some embodiments, energy use model module 314 uses a partial leastsquares regression (PLSR) method. In other embodiments, energy use modelmodule 314 may use other methods, such as ridge regression (RR),principal component regression (PCR), weighted least squares regression(WLSR), or ordinary least squares regression (OLSR).

Generally, a least squares estimation problem can be stated as follows:given a linear model

Y=Xβ+e, e˜N(0,t _(k)σ²),

find the vector {circumflex over (β)} that minimizes the sum of squarederror RSS, where

${RSS} = {{\sum\limits_{k = 1}^{n}{\frac{{\hat{e}}_{k}^{2}}{t_{k}}\mspace{14mu} {and}\mspace{14mu} \hat{e}}} = {{{Y - {X\; \hat{\beta}}}}.}}$

The optimal value of {circumflex over (β)} based on a least squaresestimation has the solution

{circumflex over (β)}=(X ^(T) X)⁻¹ X ^(T) Y.

Given n observations of the dependent variable Y (e.g., building energyconsumption) and the independent predictor variables X, energy use modelmodule 314 may estimate the regression coefficient vector {circumflexover (β)}. The estimates of the regression coefficients {circumflex over(β)} may have an asymptotic normal distribution such that ({circumflexover (β)}−β)˜AsN(0, P_(β)), where P_(β) is a covariance matrix of{circumflex over (β)} having a size of p+1 by p+1.

Still referring to FIG. 3, memory 308 is shown to include a vectoraugmentation module 316. Vector augmentation module 316 may beconfigured to augment (e.g., modify, add to, supplement, etc.) thevector of regression model coefficient estimates {circumflex over (β)}to include one or more balance point parameters (e.g., a cooling balancepoint parameter T_(bC) and/or a heating balance point parameter T_(bH)).For example, vector augmentation module 316 may combine the vector ofregression model coefficient estimates {circumflex over (β)} with one ormore balance point parameter estimates {circumflex over (T)}_(bC) and/or{circumflex over (T)}_(bH) to create vector of parameter estimates{circumflex over (θ)}(e.g., {circumflex over (θ)}=[{circumflex over (β)}{circumflex over (T)}_(bC) {circumflex over (T)}_(bH)]^(T)).

Advantageously, by combining the regression model coefficient estimates{circumflex over (β)} and the balance point parameter estimates{circumflex over (T)}_(bC) and {circumflex over (T)}_(bH) into a singleparameter vector {circumflex over (θ)}, the values for the regressionmodel coefficients {circumflex over (β)} and the balance pointparameters {circumflex over (T)}_(bC) and {circumflex over (T)}_(bH) maybe estimated together using a single regression process.

In some embodiments, one or more of the predictor variables x₁ . . .x_(p) in the predictor variable matrix X is a function of a balancepoint parameter. For example, predictor variables x₁ and x₂ maycorrespond to cooling degree days (CDD) and heating degree days (HDD),respectively, and can be determined using the following equations:

x_(1, k) = f(T̂_(bC)) = ∫_(t_(k))max (0, T_(OA)(t) − T̂_(bC)) tx_(2, k) = f(T̂_(bH)) = ∫_(t_(k))max (0, T̂_(bH) − T_(OA)(t)) t

These equations may be used to complete the predictor variable matrix Xand to facilitate the simultaneous estimation of the regression modelcoefficients {circumflex over (β)} and the balance point parameters{circumflex over (T)}_(bC) and {circumflex over (T)}_(bH).

Still referring to FIG. 3, memory 308 is shown to include an outputgradient module 318. Output gradient module 318 may be configured tocalculate a gradient ψ(k,{circumflex over (θ)}) of the output of theenergy use model with respect to the model parameters {circumflex over(θ)}. The output of the energy use model may be energy consumptionŷ_(k)=β₀t_(k)+β₁x_(1,k)+β₂ x_(2,k)+ . . . +β_(p)x_(p,k)+e_(k). Outputgradient module 318 may calculate the gradient ψ(k,{circumflex over(θ)}) according to the following equation:

${{\psi ( {k,\hat{\theta}} )} = \lbrack {\frac{\partial{\hat{y}}_{k}}{\partial\beta_{0}}\mspace{14mu} \ldots \mspace{14mu} \frac{\partial{\hat{y}}_{k}}{\partial\beta_{p}}\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}\frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}} \rbrack_{\theta = \hat{\theta}}^{T}},$

where the model parameters θ are evaluated at their estimated values{circumflex over (θ)}, as indicated by the hat notation (e.g.,{circumflex over (θ)}=[{circumflex over (β)} {circumflex over (T)}_(bC){circumflex over (T)}_(bH)]^(T)). Gradient ψ(k,{circumflex over (θ)})may be a vector having a size of p+3 (i.e., p+1 regression modelcoefficient β terms and two balance point terms).

The terms of output gradient ψ(k, {circumflex over (θ)}) with respect tothe cooling balance point T_(bC)

$( {{i.e.},\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}} )$

and heating balance point T_(bH)

$( {i.e.\mspace{11mu} \frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}} )$

can be evaluated using the relationship between predictor variable x₁and balance point T_(bC) and the relationship between predictor variablex₂ and balance point T_(bH). For example, terms

$\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}$

can be evaluated using the following equations:

$\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}} = {{\frac{\partial{\hat{y}}_{k}}{\partial x_{1,k}}\frac{\partial x_{1,k}}{\partial T_{bC}}} = {{- \beta_{1}}t_{k}^{\prime}}}$${\frac{\partial{\hat{y}}_{k}}{\partial T_{bH}} = {{\frac{\partial{\hat{y}}_{k}}{\partial x_{2,k}}\frac{\partial x_{2,k}}{\partial T_{bH}}} = {{- \beta_{2}}t_{k}^{''}}}},$

where the variable t′_(k) corresponds to the total time duringobservation period t_(k) during which T_(OA)>T_(bC), and where thevariable t″_(k) corresponds to the total time during observation periodt_(k) during which T_(OA)<T_(bH). Output gradient module 318 maycalculate the values for each of the terms in gradient ψ(k,{circumflexover (θ)}) and store gradient ψ(k,{circumflex over (θ)}) in memory.

Still referring to FIG. 3, memory 308 is shown to include a covariancematrix module 320. Covariance matrix module 320 may be configured todetermine a covariance matrix {circumflex over (P)}_(θ)using thegradient ψ(k,{circumflex over (θ)}). In some embodiments, covariancematrix module 320 may calculate covariance matrix {circumflex over(P)}_(θ) according to the following equation:

${{\hat{P}}_{\theta} = {{\hat{\sigma}}_{e}^{2}\lbrack {\sum\limits_{k = 1}^{n}{{\psi ( {k,\hat{\theta}} )}{\psi^{T}( {k,\hat{\theta}} )}}} \rbrack}^{- 1}},$

where ψ(k,{circumflex over (θ)}) is the output gradient calculated byoutput gradient module 318 and where {circumflex over (σ)}_(e), is theestimated variance of the model error

$( {{e.g.},{{\hat{\sigma}}_{e}^{2} = \frac{e^{T}e}{n - p - 1}}} ).$

Covariance matrix module 320 may generate a covariance matrix{circumflex over (P)}_(θ) having a size of p+3 by p+3.

Still referring to FIG. 3, memory 308 is shown to include an uncertaintydetermination module 322. Uncertainty determination module 322 may beconfigured to identify an uncertainty of one or more of the modelparameter estimates {circumflex over (θ)}. In some embodiments,uncertainty determination module 322 identifies the uncertainty in oneor more of the model parameters in vector {circumflex over (θ)} usingcovariance matrix {circumflex over (P)}_(θ). The uncertainty of thei^(th) parameter in vector {circumflex over (θ)} may correspond to thei^(th) entry along the diagonal (i.e., from top left to bottom right) ofcovariance matrix {circumflex over (P)}_(θ). For example, theuncertainty of regression model coefficient β₀ (i.e., the first term inparameter vector {circumflex over (θ)}) may correspond to the top leftentry of covariance matrix {circumflex over (P)}_(θ) (i.e., the firstentry along the diagonal from top left to bottom right). The uncertaintyof balance points T_(bC) and T_(bH) (i.e., terms p+2 and p+3 inparameter vector {circumflex over (θ)}) may correspond to entries p+2and p+3, respectively, along the diagonal in covariance matrix{circumflex over (P)}_(θ).

Advantageously, uncertainty determination module 322 may use thecovariance matrix {circumflex over (P)}_(θ) to determine theuncertainties in both the regression model parameters {circumflex over(β)} and the balance point parameters {circumflex over (T)}_(bC) and{circumflex over (T)}_(bH). This advantage provides improved dataanalysis functionality over traditional methods which estimate theuncertainties for only the regression model coefficients {circumflexover (β)} (and not the balance point parameters). Additionally,simultaneous estimation of the uncertainties in both the balance pointparameters {circumflex over (T)}_(bC) and/or {circumflex over (T)}_(bH)and the regression model coefficients {circumflex over (β)} may improvethe certainty of any calculations which use the estimated modelparameters {circumflex over (θ)}. This improved certainty strengthensany conclusions based on the values and/or uncertainties of the modelparameters {circumflex over (θ)}(e.g., in a LEAN energy analysis).

In some embodiments, building analysis system 110 estimates theuncertainties in model parameters {circumflex over (θ)} using thetechnique described with reference to memory modules 316-322. Thistechnique may be referred to as the “analytical technique.” In briefreview, the analytical technique may include using vector augmentationmodule 316 to augment the vector of regression coefficients {circumflexover (β)} with one or more balance points (e.g., {circumflex over(T)}_(bC) and {circumflex over (T)}_(bH)) to produce a vector of modelparameters {circumflex over (θ)}. Output gradient module 318 may use thevector of model parameters {circumflex over (θ)} to calculate a gradientof the model output ψ(k,{circumflex over (θ)}) with respect to the modelparameters. Covariance matrix module 320 may then calculate a covariancematrix {circumflex over (P)}_(θ) using the gradient of the model outputψ(k,{circumflex over (θ)}). Uncertainty determination module 322 maydetermine the uncertainties in the model parameters {circumflex over(θ)} by inspecting the covariance matrix {circumflex over (P)}_(θ).

In other embodiments, building analysis system 110 estimates theuncertainties in model parameters {circumflex over (θ)} using anempirical estimation technique. The empirical estimation technique mayutilize a re-sampling method (e.g., a Bootstrap method, a Jackknifemethod, etc.) to generate multiple samples from the set of buildingperformance data stored in building data module 310 and predictorvariable module 312. For each of the multiple samples, an estimate ofthe model parameters {circumflex over (θ)} may be determined. Theuncertainty in the model parameters {circumflex over (θ)}can then becalculated using the multiple estimates of the model parameters{circumflex over (θ)}. The empirical estimation technique is explainedin greater detail with reference to data sampling module 324 andempirical analysis module 326.

Still referring to FIG. 3, memory 308 is shown to include a datasampling module 324. Data sampling module 324 may receive a set ofbuilding performance data including values for one or more independentvariables (e.g., predictor variables x_(1,k) . . . x_(p,k), time periodduration t_(k), etc.) and an associated energy consumption value y_(k)for the building site. The set of building performance data may includen data points, each data point corresponding to an observation of theindependent variable y_(k) and the dependent variables x_(1,k) . . .x_(p,k) and t_(k), where k=1 . . . n.

In some embodiments, the set of data points may be represented by apredictor variable matrix X and an energy consumption vector Y. Thepredictor variable matrix X may be a n by p+1 matrix includingcalculated or measured values for each of p predictor variables (i.e.,x_(1,k) . . . x_(p,k)) and a time variable t_(k). The value of timevariable t_(k) may identify a time duration associated with eachobservation period k (e.g., number of hours or days per period, etc.).The predictor variable values and/or predictor variable matrix may bestored in predictor variable module 312. The energy consumption vector Ymay include an energy consumption value y_(k) associated with eachobservation period k, where k=1 . . . n. Each row of the predictorvariable matrix X and energy consumption vector Y may correspond to adifferent data point or observation of the dependent and independentvariables.

Data sampling module 324 may generate multiple samples from the set ofdata points. Each of the multiple samples may include a plurality ofdata points selected from the set of data points. In some embodiments,data sampling module 324 uses a Bootstrap sampling process to generatethe multiple samples. Using the Bootstrap sampling process, datasampling module 324 may replace a predetermined number of data points inthe set of data points with an equal number of data points selectedrandomly from the original set of data points (i.e., sampling withreplacement). The Bootstrap sampling process may produce a sample havingthe same number of data points as the original set (e.g., n datapoints), with some of the data points potentially repeated multipletimes. Data sampling module 324 may perform the Bootstrap samplingprocess multiple times to generate multiple samples from the set of datapoints.

In some embodiments, data sampling module 324 uses a Jackknife samplingprocess to generate the multiple samples. Using the Jackknife samplingprocess, data sampling module 324 may remove one of the data points fromthe original set of n data points to generate a sample of n−1 datapoints. Data sampling module 324 may perform the Jackknife process ntimes, removing a different data point each time. Using the Jackknifesampling process data sampling module 324 may generate n samples of n−1data points. The i^(th) Jackknife sample may include all of the datapoints in the original set of data points, except for the i^(th) datapoint, where i=1 . . . n. Both the Bootstrap and the Jackknife methodmay be similar to the Monte Carlo method. However, with the Bootstrapand Jackknife methods, there is no assumption of any particulardistribution.

Still referring to FIG. 3, memory 308 is shown to include an empiricalanalysis module 326. Empirical analysis module 326 may be configured toestimate the model parameters {circumflex over (θ)} for each of themultiple samples using the plurality of data points included in thesample. In some embodiments, empirical analysis module 326 estimates themodel parameters {circumflex over (θ)} in multiple stages. For example,empirical analysis module 326 may first estimate the balance point orpoints for a particular data sample. Then, empirical analysis module 326may use the estimated balance point(s) to calculate any predictorvariable values that depend on a balance point (e.g., CDD and/or HDDvalues) and to determine the remaining building parameters (e.g.,regression model coefficients {circumflex over (β)}).

In some embodiments, empirical analysis module 326 uses outside airtemperature and energy consumption data to estimate the balance points(e.g., {circumflex over (T)}_(bC) and/or {circumflex over (T)}_(bH)).For example, empirical analysis module 326 may analyze the energyconsumption and outside air temperature data to identify a maximumoutside air temperature (i.e., for heating balance point {circumflexover (T)}_(bH)) or a minimum outside air temperature (i.e., for coolingbalance point {circumflex over (T)}_(bC)) below or above which buildingenergy consumption is a function of the outside air temperature.

According to another embodiment, empirical analysis module 326 uses anoptimization scheme to determine the balance point or points. Theoptimization scheme may include an exhaustive search of the balancepoints, a gradient descent algorithm, and/or a generalized reducedgradient (GRG) method to estimate balance points {circumflex over(T)}_(bC) and/or {circumflex over (T)}_(bH). In some embodiments,empirical analysis module 326 identifies the balance points using aniteratively reweighted least squares regression method. For example,empirical analysis module 326 may search for balance points thatminimize the sum of squared error RSS of the building energy use model,where

${RSS} = {{\sum\limits_{i = 1}^{n}\; {\frac{{\hat{e}}_{i}^{2}}{t_{i}}\mspace{14mu} {and}\mspace{14mu} \hat{e}}} = {{{Y - {X\; \hat{\beta}}}}.}}$

Empirical analysis module 326 may estimate balance point values for eachof the multiple samples generated by data sampling module 324.

In some embodiments, empirical analysis module 326 uses the estimatedbalance points {circumflex over (T)}_(bC) and/or {circumflex over(T)}_(bH) to calculate values for any predictor variables that depend ona balance point (e.g., CDD, HDD, CED, HED, etc.). For example, ifpredictor variable x_(1,k) corresponds to cooling degree days andpredictor variable x_(2,k) corresponds to heating degree days, empiricalanalysis module 326 may calculate values for predictor variables x_(1,k)and x_(2,k) using the following equations:

x_(1, k) = f(T̂_(bC)) = ∫_(t_(k))max (0, T_(OA)(t) − T̂_(bC)) tx_(2, k) = f(T̂_(bH)) = ∫_(t_(k))max (0, T̂_(bH) − T_(OA)(t)) t

Empirical analysis module 326 may use the predictor variable valuesx_(1,k) . . . x_(p,k) and the associated energy consumption values y_(k)to calculate the regression model coefficients {circumflex over (β)}.Empirical analysis module 326 may use any of a variety of differentestimation techniques to estimate regression model coefficients{circumflex over (β)}. In some embodiments, empirical analysis module326 uses a partial least squares regression (PLSR) method. In otherembodiments, empirical analysis module 326 may use other methods, suchas ridge regression (RR), principal component regression (PCR), weightedleast squares regression (WLSR), or ordinary least squares regression(OLSR). The optimal value of {circumflex over (β)} based on a leastsquares estimation has the solution {circumflex over(β)}=(X^(T)X)⁻¹X^(T)Y.

In some embodiments, empirical analysis module 326 uses the estimatedbalance points for each of the multiple samples to calculate anuncertainty in the balance point estimates. For example, the uncertaintyin a balance point estimate {circumflex over (T)}_(b) (i.e., {circumflexover (T)}_(bC) or {circumflex over (T)}_(bH)) can be observed byexamining the mean and standard deviation of the set of balance pointestimates obtained from the multiple samples generated by data samplingmodule 324. The mean T_(b) and standard deviation σ_(T) _(b) of the setof balance point estimates {circumflex over (T)}_(b,1) . . . {circumflexover (T)}_(b,N) can be determined using the following equations:

${\overset{\_}{T}}_{b} = {\frac{1}{N}{\sum\limits_{j = 1}^{N}\; {\hat{T}}_{b,j}}}$${\sigma_{T_{b}} = \sqrt{\frac{1}{N}{\sum\limits_{j = 1}^{N}\; ( {{\hat{T}}_{b,j} - {\overset{\_}{T}}_{b}} )^{2}}}},$

where N is the number of samples generated by data sampling module 324and {circumflex over (T)}_(b,j) is the j^(th) balance point estimate,j=1 . . . N.

In some embodiments, empirical analysis module 326 calculates theuncertainties in the estimated values of the regression modelcoefficients {circumflex over (β)} and the uncertainties in theestimated values of the balance points {circumflex over (T)}_(bC) and/or{circumflex over (T)}_(bH) simultaneously. Advantageously,simultaneously calculating the uncertainties in the regression modelcoefficients estimates and the balance point estimates may allow for amultidimensional uncertainty to be determined by empirical analysismodule 326 for use by energy analysis module 328.

Still referring to FIG. 3, memory 308 is shown to include an energyanalysis module 328. Energy analysis module 328 may be configured toanalyze a building's energy performance using a building energy usemodel. Energy analysis module 328 may perform a variety of energyanalysis functions including, for example, generating energy savingsestimates, detecting outlier building sites with poor energy performance(e.g., by comparing similar building sites), detecting faults, anddetermining the effects of a fault on a building's energy consumption.

In some embodiments, energy analysis module 328 may perform energyanalysis using a minimal amount of building performance data (e.g., aLEAN energy analysis). Energy analysis module 328 may rely on the valuesand corresponding uncertainties of parameters in the building energy usemodel (e.g., the parameters in parameter vector {circumflex over (θ)})to arrive at conclusions regarding a building's energy performance.

In some embodiments, energy analysis module 328 is configured to performoutlier detection. Energy analysis module 328 may be configured tocompare one or more statistics of a test building to the probabilitydistribution of those statistics for the other buildings in the sameclass (e.g., buildings having similar usage characteristics, buildingslocated in similar geographic regions, etc.). For example, energyanalysis module 328 may determine that a building's statistic is anoutlier for the class based on a number of standard deviations that thestatistic is above or below the mean for the class distribution. Invarious embodiments, energy analysis module 328 may use any number ofoutlier detection techniques to identify an outlier value. For example,energy analysis module 328 may use a generalized extreme studentizeddeviate test (GESD), Grubb's test, or any other form of univariateoutlier detection technique. In some embodiments, energy analysis module328 may identify a building as an outlier if the statistic for thebuilding is within a fixed percentage of the minimum or maximum for theclass distribution (e.g., top 5%, bottom 5%, top 10%, etc.).

In some embodiments energy analysis module 328 may use a distance valuebetween statistics to detect an outlier. For example, energy analysismodule 328 may determine a Mahalanobis distance to compare statistics.Such a distance may represent a statistical distance away from thetypical building in the class. If the Mahalanobis distance for a testbuilding is above a critical value, energy analysis module 328 maygenerate an indication that the building's one or more statistics areoutliers in relation to the other buildings in the class. In someembodiments, energy analysis module 328 may project the distance ontothe vector directions defining changes in a building's parameters todetermine a root cause of the change. Other outlier detection techniquesthat may be used by energy analysis module 328 include, but are notlimited to, Wilkes' method (e.g., if multivariate analysis is used) andvarious cluster analysis techniques.

Energy analysis module 328 may be configured to detect excessive energyconsumption by a building. In some embodiments, energy analysis module328 may perform one or more hypothesis tests using the building datastored in building data module 310 and the model parameters {circumflexover (θ)} to detect excessive energy consumption. Exemplary hypothesistests include F-tests and Chi-squared tests. In some embodiments,hypothesis testing may be used to test one or more values against abaseline, as described in U.S. patent application Ser. No. 13/252,092entitled “Systems and Methods for Detecting Changes in Energy Usage In aBuilding” and filed on Oct. 3, 2011, the entirety of which isincorporated by reference herein.

In some embodiments, energy analysis module 328 may be configured todetect faults and to determine the effects of a fault on a building'senergy consumption. In various embodiments, energy analysis module 328may determine one or more changes to the model parameters of the energyuse model (e.g., changes to the parameters in parameter vector{circumflex over (θ)}) that result when a particular fault is present.For example, energy analysis module 328 may determine changes to the setof vector parameters {circumflex over (θ)} that result from a damperbeing stuck in the open position. In one embodiment, energy analysismodule 328 uses a simulation model to determine the changes to theenergy use model parameters. In another embodiment, energy analysismodule 328 determines a mapping between changes to a building's energyuse model parameters and its physical parameters (e.g., the building'scooling slope S_(C), heating slope S_(H), cooling balance point T_(bC),etc.).

Energy analysis module 328 may provide the changes to the energy usemodel parameters to energy use model module 314. Energy use model module314 may then determine a corresponding change to the building's energyconsumption. For example, a stuck damper of an AHU may cause abuilding's normalized annual energy consumption to increase by 25,000kWh/year. Energy analysis module 328 may use this change in energyconsumption to calculate a corresponding financial cost associated withthe fault condition. For example, energy analysis module 328 maymultiply the determined change in energy consumption by a price per unitenergy (e.g., received from utility 114) to calculate a financial costassociated with the fault.

In some embodiments, energy analysis module 328 may be configured toperform fault detection and analysis of the building under study usingthe energy use model generated by energy use model module 314. In oneembodiment, energy analysis module 328 may monitor changes to thebuilding's energy use model's parameters over time to detect potentialfaults. In another embodiment, energy analysis module 328 may performfault detection using peer analysis with other buildings in its class todetect potential faults. For example, buildings having outlier modelparameter changes may be identified as having potential faults. If apotential fault is detected, energy analysis module 328 may use amapping between energy use model parameters and the building's physicalparameters to determine a cause of the fault.

Advantageously, the various energy analysis processes performed byenergy analysis module 328 may be improved (e.g., improved accuracy,improved certainty, etc.) by calculating the uncertainties in the modelparameters {circumflex over (θ)} using the systems and methods of thepresent disclosure.

Still referring to FIG. 3, memory 308 is shown to include an output andclient request module 330. Output and client request module 330 may beconfigured to process user input received via communications interface302 and/or user interface I/O 303. For example, output and clientrequest module 330 may process a user request for the value of a modelparameter and/or the uncertainty in a model parameter. As anotherexample, output and client request module 330 may process a user requestto run a performance analysis and/or generate an analytical performanceanalysis report for a particular building or building system. Output andclient request module 330 may be configured to run or query uncertaintydetermination module 322, energy use model module 314, energy analysismodule 328, or any other component of building analysis system 110 todetermine a response to the user request.

Output and client request module 330 may be configured to generate anoutput for presentation to a user. Output and client request module 330may generate a graphical display, a visual display, a textual display,or any other type of user-comprehensible output. Output and clientrequest module 330 may communicate a result of a user query/request(e.g., an appropriate parameter order of a particular building energyuse model, an analytical report, etc.), a result of an intermediateprocessing step (e.g., a test statistic or regression statistic value,etc.), a result of a performance analysis, a result of a fault detectionanalysis, or any other data stored or used by building analysis system110. In various embodiments, output and client request module 330 maygenerate display data for presentation via a local display (e.g., to alocal user interacting with building analysis system 110 via userinterface I/O 303), or may communicate output data to a remote user viacommunications interface 302 (e.g., a user interacting with buildinganalysis system 110 via a network connection and/or a remote client).

Still referring to FIG. 3, memory 308 is shown to include a buildingcontrol services module 332. Building control services module 332 may beconfigured to control one or more buildings, building systems, orbuilding subsystems using a building energy use model. For example,building control services module 332 may utilize closed loop control,feedback control, PI control, model predictive control, or any othertype of automated building control methodology that relies on a model totranslate an input into an output. In some embodiments, building controlservices module 332 uses the building energy use model maintained byenergy use model module 314 along with the uncertainties of variousmodel parameters to translate an input received from a building systeminto an output or control signal for the building system.

Building control services module 332 may receive inputs from sensorydevices (e.g., temperature sensors, pressure sensors, flow rate sensors,humidity sensors, electric current sensors, cameras, radio frequencysensors, microphones, etc.), user input devices (e.g., computerterminals, client devices, user devices, etc.) or other data inputdevices via communications interface 302 and/or user interface I/O 303.Building control services module 332 may apply the various inputs to abuilding energy use model to determine an output for one or morebuilding control devices (e.g., dampers, air handling units, chillers,boilers, fans, pumps, etc.) in order to affect a variable state orcondition within a building or building system associated with thebuilding energy use model (e.g., zone temperature, humidity, air flowrate, etc.). Building control services module 332 may operate thebuilding or building system to maintain building conditions (e.g.,temperature, humidity, air quality, etc.) within a setpoint range, tooptimize energy performance (e.g., to minimize energy consumption, tominimize energy cost, etc.), and/or to satisfy any constraint orcombination of constraints as may be desirable for variousimplementations.

Referring now to FIG. 4, a flowchart of a process 400 for determiningthe uncertainty in parameters of a building energy use model is shown,according to an exemplary embodiment. In some embodiments, process 400is performed by building analysis system 110 using processing circuit304. Process 400 illustrates an analytical process for determining theuncertainties in a vector {circumflex over (θ)} of energy use modelparameters. Vector {circumflex over (θ)} may include estimates for oneor more balance point parameters (i.e., a cooling balance pointparameter {circumflex over (T)}_(bC) and/or a heating balance pointparameter {circumflex over (T)}_(bH)) and one or more coefficients of abuilding energy use model (e.g., regression model coefficients β).Advantageously, process 400 may be used to estimate both the values andthe uncertainties of the balance point parameters (e.g., {circumflexover (T)}_(bC) and/or {circumflex over (T)}_(bH)) and the regressionmodel coefficients β. This advantage provides improved data analysisfunctionality over traditional methods which estimate the uncertaintiesfor only the regression model coefficients {circumflex over (β)} (andnot the balance point parameters).

In some embodiments, process 400 estimates the uncertainties in thevalues of the balance point parameters {circumflex over (T)}_(bC) and/or{circumflex over (T)}_(bH) and the uncertainties in the values of theregression model coefficients {circumflex over (β)} simultaneously.Advantageously, simultaneous estimation of the uncertainties in both thebalance point parameters {circumflex over (T)}_(bC) and/or {circumflexover (T)}_(bH) and the regression model coefficients β may improve theuncertainties in all of the model parameters {circumflex over (θ)} overtraditional uncertainty estimation techniques. The improved uncertaintystrengthens any conclusions based on the values and/or uncertainties ofthe model parameters {circumflex over (θ)}(e.g., in a LEAN energyanalysis).

Still referring to FIG. 4, process 400 is shown to include receiving anenergy use model for a building site (step 402). The energy use modelmay be of any form including parametric models (e.g., linear regressionmodels, non-linear regression models, etc.), non-parametric models(neural networks, kernel estimation, hierarchical Bayesian, etc.), orsomething in between (e.g., Gaussian process models). In someembodiments, step 402 includes receiving an energy use model from anexternal source (e.g., via communications interface 302). In otherembodiments, step 402 includes generating the energy use model (e.g.,using the building data stored in building data module 310 and/or thepredictor variable data stored in predictor variable module 312).

In some embodiments, linear regression is used in step 402 to generatethe energy use model. The energy use model may include one or morepredictor variables and one or more model parameters. A linearregression model for a building may be represented by the followingequation:

Y=Xβ+e,

where Y is the building's energy consumption, X is a predictor variablematrix, β is a vector of unknown regression coefficients (e.g.,)β₀β₁, .. . β_(p)), and e is the model error such that e˜N(0,t_(k)σ²).

The predictor variable matrix X may have a size of n by p+1 where n isthe total number of observations and where the number p+1 includes ppredictor variables (e.g., x₁ . . . x_(p)) and a time variable trepresenting the duration of each observation period (e.g., t=[t₁ . . .t_(n)]^(T)). For example, the estimated energy consumption during thek^(th) observation period can be expressed as:

ŷ _(k)=β₀ t _(k)+β₁ x _(1,k)+β₂ x _(2,k)+ . . . +β_(p) x _(p,k) +e _(k),

where t_(k) is the time duration of the k^(th) observation period k=1 .. . n.

In some embodiments, the energy use model includes a weather-relatedpredictor variable (e.g., outside air temperature T_(OA), enthalpy,cooling degree days, heating degree days, heating energy days, coolingenergy days, etc.). In some embodiments, the energy use model includesonly one weather-related predictor variable. In some embodiments, theenergy use model includes one or more non-weather-related predictorvariables. Non-weather-related predictor variables may include, forexample, water consumption, building occupancy, days off, the number ofdays per period t, and/or any other variable which may affect the site'senergy consumption. The weather-related predictor variable and/or otherpredictor variables may be included in predictor variable matrix X.

In some embodiments, step 402 includes augmenting (e.g., modify, add to,supplement, etc.) the vector of regression model coefficient estimates{circumflex over (β)} with one or more balance point parameters (i.e.,{circumflex over (T)}_(bC) and/or {circumflex over (T)}_(bH)) to createvector of parameter estimates {circumflex over (θ)}(e.g., {circumflexover (θ)}=[{circumflex over (β)} {circumflex over (T)}_(bC) {circumflexover (T)}_(bH)]^(T)). Advantageously, by combining the regression modelcoefficient estimates) {circumflex over (β)} and the balance pointparameter estimates {circumflex over (T)}_(bC) and {circumflex over(T)}_(bH) into a single parameter vector {circumflex over (θ)}, theuncertainties in the values for the regression model coefficients{circumflex over (β)} and the uncertainties in the values for thebalance point parameters {circumflex over (T)}_(bC) and {circumflex over(T)}_(bH) may be estimated simultaneously using a single regressionprocess.

Still referring to FIG. 4, process 400 is shown to include calculating agradient of an output of the energy use model with respect to the modelparameters (step 404). The gradient of the output may be the gradientψ(k,{circumflex over (θ)}) of the energy consumption ŷ_(k) with respectto the model parameters {circumflex over (θ)}, whereŷ_(k)=β₀t_(k)+β₁x_(1,k)+β₂x_(2,k)+ . . . +β_(p)x_(p,k)+e_(k). In someembodiments, step 404 is performed by output gradient module 318 asdescribed with reference to FIG. 3.

In step 404, the gradient ψ(k,{circumflex over (θ)}) of the model outputŷ_(k) may be calculated using the following equation:

${{\psi( {k,\hat{\theta}} )} = \begin{bmatrix}\frac{\partial{\hat{y}}_{k}}{\partial\beta_{0}} & \ldots & \frac{\partial{\hat{y}}_{k}}{\partial\beta_{p}} & \frac{\partial{\hat{y}}_{k}}{\partial T_{bC}} & \frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}\end{bmatrix}_{\theta = \hat{\theta}}^{T}},$

where the model parameters θ are evaluated at their estimated values{circumflex over (θ)}, as indicated by the hat notation (e.g.,{circumflex over (θ)}=[{circumflex over (β)} {circumflex over (T)}_(bC){circumflex over (T)}_(bH)]^(T)). Gradient ψ(k,{circumflex over (θ)})may be a vector having a size of p+3 (i.e., p+1 regression coefficientβterms and two balance point terms).

In some embodiments, step 404 includes evaluating and/or simplifying theterms of output gradient ψ(k,{circumflex over (θ)}). In someembodiments, one or more of the predictor variables x₁ . . . x_(p) inthe predictor variable matrix X may be a function of a balance pointparameter. For example, predictor variables x₁ and x₂ may correspond tocooling degree days (CDD) and heating degree days (HDD), respectively,and can be determined using the following equations:

x_(1, k) = f(T̂_(bC)) = ∫_(t_(k))max (0, T_(OA)(t) − T̂_(bC)) tx_(2, k) = f(T̂_(bH)) = ∫_(t_(k))max (0, T̂_(bH) − T_(OA)(t)) t

The terms of output gradient ψ(k,{circumflex over (θ)}) with respect tothe cooling balance point T_(bC)

$( {{i.e.},\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}} )$

and heating balance point T_(bH)

$( {{i.e.},\frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}} )$

can be evaluated using the relationship between predictor variable x₁and balance point T_(bC) and the relationship between predictor variablex₂ and balance point T_(bH). For example, terms

$\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}$

can be evaluated using the following equations:

$\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}} = {{\frac{\partial{\hat{y}}_{k}}{\partial x_{1,k}}\frac{\partial x_{1,k}}{\partial T_{bC}}} = {{- \beta_{1}}t_{k}^{\prime}}}$${\frac{\partial{\hat{y}}_{k}}{\partial T_{bH}} = {{\frac{\partial{\hat{y}}_{k}}{\partial x_{2,k}}\frac{\partial x_{2,k}}{\partial T_{bH}}} = {\beta_{2}t_{k}^{''}}}},$

where the variable t′_(k) corresponds to the total time duringobservation period t_(k) during T_(OA)>T_(bC), and where the variablet″_(k) corresponds to the total time during observation period t_(k)during which T_(OA)<T_(bH). Step 404 may include substituting the values−β₁t′_(k) and β₂t″_(k) for terms

${\frac{\partial{\hat{y}}_{k}}{\partial T_{bC}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial{\hat{y}}_{k}}{\partial T_{bH}}},$

respectively.

Still referring to FIG. 4, process 400 is shown to include determining acovariance matrix {circumflex over (P)}_(θ) using the calculatedgradient ψ(k,{circumflex over (θ)}) (step 406). In some embodiments,step 406 is performed by covariance matrix module 320 as described withreference to FIG. 3. In some embodiments, step 406 includes calculatingcovariance matrix {circumflex over (P)}_(θ) according to the followingequation:

${{\hat{P}}_{\theta} = {{\hat{\sigma}}_{e}^{2}\lbrack {\sum\limits_{k = 1}^{n}\; {{\psi( {k,\hat{\theta}} )}{\psi^{T}( {k,\hat{\theta}} )}}} \rbrack}^{- 1}},$

where ψ(k,{circumflex over (θ)}) is the output gradient calculated instep 404 and where {circumflex over (σ)}_(e) is the estimated varianceof the model error

$( {{e.g.},{{\hat{\sigma}}_{e}^{2} = \frac{^{T}}{n - p - 1}}} ).$

Covariance matrix {circumflex over (P)}_(θ) may have a size of p+3 byp+3.

Still referring to FIG. 4, process 400 is shown to include using thecovariance matrix to identify the uncertainty of the model parameters(step 408). In some embodiments, step 408 is performed by uncertaintydetermination module 322 as described with reference to FIG. 3. In someembodiments, step 408 includes identifying the uncertainty in one ormore of the model parameters in vector {circumflex over (θ)} usingcovariance matrix {circumflex over (P)}_(θ). The uncertainty of thei^(th) parameter in vector {circumflex over (θ)} may correspond to thei^(th) entry along the diagonal (i.e., from top left to bottom right) ofcovariance matrix {circumflex over (P)}_(θ). For example, theuncertainty of regression model coefficient β₀ (i.e., the first term inparameter vector {circumflex over (θ)}) may correspond to the top leftentry of covariance matrix {circumflex over (P)}_(θ) (i.e., the firstentry along the diagonal from top left to bottom right). The uncertaintyof balance points T_(bC) and T_(bH) (i.e., terms p+2 and p+3 inparameter vector {circumflex over (θ)}) may correspond to entries p+2and p+3, respectively, along the diagonal in covariance matrix{circumflex over (P)}_(θ).

Advantageously, step 408 may be performed to determine the uncertaintiesin both the regression model parameters {circumflex over (β)} and thebalance point parameters {circumflex over (T)}_(bC) and/or {circumflexover (T)}_(bH). The uncertainties in both the regression modelparameters and the balance point parameters may be calculatedsimultaneously in step 406 (i.e., by determining covariance matrix{circumflex over (P)}_(θ)) and subsequently identified in step 408 asentries in the covariance matrix {circumflex over (P)}_(θ). Thisadvantage provides improved data analysis functionality over traditionalmethods which estimate the uncertainties for only the regression modelcoefficients {circumflex over (β)} (and not the balance pointparameters). This improvement strengthens any conclusions based on thevalues and/or uncertainties of the model parameters {circumflex over(θ)} (e.g., in a LEAN energy analysis).

In some embodiments, process 400 includes performing a multivariateuncertainty analysis of two or more model parameters in {circumflex over(θ)} (e.g., a balance point parameter and a regression modelcoefficient). The regression model coefficient included in themultivariate uncertainty analysis may be associated with aweather-related predictor variable that is a function of the balancepoint parameter. The multivariate uncertainty analysis may serve as atool for visualizing (e.g., identifying, determining, graphicallyrepresenting, etc.) a correlation between model parameters (assuming aprobability distribution of the balance point parameter and thecorresponding regression model coefficient). For example, a correlationbetween model parameters may be identified and/or visualized in agraphical display (e.g., the graph illustrated in FIG. 2, a visualoutput of the model parameters or the uncertainties associatedtherewith, an energy consumption visualization based on the modelparameters, etc.). Such a multivariate uncertainty analysis is notpossible using traditional regression techniques which only calculatethe uncertainties in the regression model coefficients {circumflex over(β)}.

In some embodiments, process 400 includes performing a multivariateoutlier peer analysis. The multivariate outlier peer analysis may serveas a tool for visualizing (e.g., identifying, determining, graphicallyrepresenting, etc.) a correlation between model parameters of multipleenergy use models (e.g., for multiple building sites). Such amultivariate outlier peer analysis is made possible by the calculationof the uncertainties in the model parameters {circumflex over (θ)} andis not possible using traditional regression techniques which onlycalculate the uncertainties in the regression model coefficients{circumflex over (β)}.

Still referring to FIG. 4, process 400 is shown to include updating theenergy use model using the uncertainty of the model parameters (step410). In various embodiments, step 410 may include updating one or moreregression coefficient values, updating one or more balance pointvalues, or updating the uncertainties associated with the regressioncoefficient values and/or balance point values.

Step 410 may include updating the current energy use model to reflectthe most recent data received from the building system. For example,step 410 may include using the most recent building data from buildingdata module 310 and the most recent predictor variable data frompredictor variable module 312 to update the energy use model maintainedby energy use model module 314. Step 410 may include using vectoraugmentation module 316, output gradient module 318, covariance matrixmodule 320, and/or uncertainty determination module 322 to perform aniteration of the various processing steps described with reference toFIG. 3.

In some embodiments, step 410 includes obtaining a statisticaldistribution of the estimated energy consumption predicted by the energyuse model based on the calculated covariance matrix {circumflex over(P)}_(θ) of the parameter vector {circumflex over (θ)}. Step 410 mayinclude performing a Monte Carlo simulation on the parameter vector{circumflex over (θ)}. The simulation may assume a statisticaldistribution for the parameter vector {circumflex over (θ)}. Forexample, assuming that the parameter vector {circumflex over (θ)} has aGaussian distribution with a mean {circumflex over (θ)} and a covariancematrix {circumflex over (P)}_(θ), a set of m samples can be generatedfor θ (e.g., {circumflex over (θ)}_(i) where i=1, . . . ,m). For eachsample {circumflex over (θ)}_(i), the weather-related predictorvariables in the energy use model (e.g., CDD, HDD, CED, HED, etc.) maybe updated based on the values of the balance point parameter in thecurrent sample. The energy use model can be evaluated at each parametervector sample {circumflex over (θ)}_(i) and the most recent data pointsreceived from the building system to obtain a value for the estimatedenergy consumption vector Ŷ_(i), where i=1, . . . ,m. Step 410 maygenerate a set of m energy consumption vectors which can be used todetermine the statistical distribution of the estimated energyconsumption vector Y.

Step 410 may include one or more processing steps for propagating theuncertainty calculated for {circumflex over (θ)} to an uncertainty of Ŷ.The Monte Carlo simulation described above is an example of a heuristicimplementation of such a processing step. In some embodiments, theuncertainty calculated for {circumflex over (θ)} may be propagated to anuncertainty of Ŷ analytically rather than heuristically. A statisticaldistribution for the energy consumption vector Ŷ may result in a higherconfidence in any conclusions drawn based on the estimated energyconsumption (e.g., yearly energy savings, fault detection, diagnostics,etc.).

Still referring to FIG. 4, process 400 is shown to include applyinginputs to the updated energy use model (step 412). Inputs may include,for example, inputs from sensory devices (e.g., temperature sensors,pressure sensors, flow rate sensors, humidity sensors, electric currentsensors, cameras, radio frequency sensors, microphones, etc.), userinput devices (e.g., computer terminals, client devices, user devices,etc.) or other data input devices via communications interface 302and/or user interface I/O 303. Inputs may include measured variablesindicating a current state or condition within a building or buildingsystem, setpoints, constraint conditions, control parameters, operatingschedules, or other measured, calculated, or user-defined inputs.

Step 412 may include using the updated energy use model to translate theinputs into an output or control signal for the building system. Forexample, step 412 may include using closed loop control, feedbackcontrol, PI control, model predictive control, or any other type ofautomated building control methodology that relies on a model totranslate an input into an output or control signal. In someembodiments, step 412 includes using a building energy use model thatuses balance point parameters for which an uncertainty is estimated.

Step 412 may include applying the various inputs to a building energyuse model to determine an output for one or more building controldevices (e.g., dampers, air handling units, chillers, boilers, fans,pumps, etc.) in order to affect a variable state or condition within abuilding or building system associated with the building energy usemodel (e.g., zone temperature, humidity, air flow rate, etc.). Step 412may include operating the building or building system to maintainbuilding conditions (e.g., temperature, humidity, air quality, etc.)within a setpoint range, to optimize energy performance (e.g., tominimize energy consumption, to minimize energy cost, etc.), and/or tosatisfy any constraint or combination of constraints as may be desirablefor various implementations.

Still referring to FIG. 4, process 400 is shown to include conducting aperformance analysis using the updated energy use model (step 414). Insome embodiments, step 414 is performed by energy analysis module 328 aspreviously described with reference to FIG. 3. Step 414 may includeanalyzing a building's energy performance using the updated buildingenergy use model to generate energy savings estimates, detect outlierbuilding sites with poor energy performance (e.g., by comparing similarbuilding sites), determine the effects of a fault on a building's energyconsumption, or perform other energy analysis tasks

In some embodiments, step 414 includes performing an energy analysisusing a minimal amount of building performance data (e.g., a LEAN energyanalysis). Step 414 may include using the number and value of parametersin the building energy use model (e.g., the parameters in parametervector {circumflex over (β)}) to arrive at conclusions regarding abuilding's energy performance. By calculating the uncertainty in thevarious model parameters, the accuracy of the conclusions reached instep 414 may be improved.

In some embodiments, step 414 includes performing outlier detection. Forexample, step 414 may include comparing one or more statistics of a testbuilding to the probability distribution of those statistics for theother buildings in the same class (e.g., buildings having similar usagecharacteristics, buildings located in similar geographic regions,buildings modeled by energy use models having the same number ofparameters, etc.). Step 414 may include determining that a building'sstatistic is an outlier for the class based on a number of standarddeviations that the statistic is above or below the mean for the classdistribution. In various embodiments, step 414 includes using any numberof outlier detection techniques to identify an outlier value. Forexample, step 414 may include using a generalized extreme studentizeddeviate test (GESD), Grubb's test, or any other form of univariateoutlier detection technique. In some embodiments, step 414 includesidentifying a building as an outlier if the statistic for the buildingis within a fixed percentage of the minimum or maximum for the classdistribution (e.g., top 5%, bottom 5%, top 10%, etc.).

In some embodiments step 414 includes using a distance value betweenstatistics to detect an outlier. For example, step 414 may includedetermining a Gaussian or Mahalanobis distance to compare statistics.Such a distance may represent a statistical distance away from thetypical building in the class. If the Mahalanobis distance for a testbuilding is above a critical value, step 414 may include generating anindication that the building's one or more statistics are outliers inrelation to the other buildings in the class. In some embodiments, step414 includes projecting the distance onto the vector directions definingchanges in a building's parameters to determine the root cause. Otheroutlier detection techniques that may be used in step 414 include, butare not limited to, Wilkes' method (e.g., if multivariate analysis isused) and various cluster analysis techniques.

Step 414 may include detecting excessive energy consumption by abuilding. In some embodiments, step 414 includes performing one or morehypothesis tests using the building data stored in building data module310 and the energy use model stored by energy use model module 314 todetect excessive energy consumption. Exemplary hypothesis tests includeF-tests and Chi-squared tests. In some embodiments, hypothesis testingmay be used to test one or more values against a baseline.

In some embodiments, step 414 includes determining the effects of afault on a building's energy consumption. In various embodiments, step414 includes determining one or more changes to the model parameters ofthe energy use model (e.g., changes to the parameters in parametervector β) that result when a particular fault is present. For example,step 414 may include determining changes to the vector of modelparameters β that result from a damper being stuck in the open position.In one embodiment, step 414 includes using a simulation model todetermine the changes to the energy use model parameters. In anotherembodiment, step 414 includes determining a mapping between changes to abuilding's energy use model parameters and its physical parameters(e.g., the building's cooling slope S_(C), heating slope S_(H), coolingbalance point T_(bC), etc.).

Step 414 may include providing the changes to the energy use modelparameters to energy use model module 314. Energy use model module 314may then determine a corresponding change to the building's energyconsumption. For example, a stuck damper of an AHU may cause abuilding's normalized annual energy consumption to increase by 25,000kWh/year. Energy analysis module 328 may use this change in energyconsumption to calculate a corresponding financial cost associated withthe fault condition. For example, energy analysis module 328 maymultiply the determined change in energy consumption by a price per unitenergy (e.g., received from utility 114) to calculate a financial costassociated with the fault.

In some embodiments, step 414 includes performing fault detection andanalysis of the building under study using the updated building energyuse model. In one embodiment, step 414 includes monitoring changes tothe building's energy use model's parameters over time to detectpotential faults. In another embodiment, step 414 includes performingfault detection using peer analysis with other buildings in its class todetect potential faults. For example, buildings having outlier modelparameter changes may be identified as having potential faults. If apotential fault is detected, step 414 may include using a mappingbetween energy use model parameters and the building's physicalparameters to determine the cause of the fault. Advantageously,determining the uncertainty in the model parameters (e.g., byuncertainty determination module 322) may facilitate (e.g., improve theaccuracy of) the various energy analysis functions performed in step414.

Still referring to FIG. 4, process 400 is shown to include providing anoutput using a result of the performance analysis (step 416). In someembodiments, step 416 includes generating display data for presentationto a user via a local display (e.g., to a local user interacting withbuilding analysis system 110 via user interface I/O 303). In someembodiments, step 416 includes communicating a result of the performanceanalysis to a remote user, system, or device via communicationsinterface 302 (e.g., a user interacting with building analysis system110 via a network connection and/or a remote client). Step 416 mayinclude generating an output for presentation to a user in auser-comprehensible format (e.g., visual display, graphical display,textual display, etc.) and/or storing a result of the performanceanalysis in a data storage device.

In some embodiments, step 416 includes using a result of the performanceanalysis to determine or change an output or control signal provided toa building system device. Step 416 may include providing the output forone or more building control devices (e.g., dampers, air handling units,chillers, boilers, fans, pumps, etc.) in order to affect a variablestate or condition within a building or building system associated withthe building energy use model (e.g., zone temperature, humidity, airflow rate, etc.). Step 416 may include operating the building orbuilding system to maintain building conditions (e.g., temperature,humidity, air quality, etc.) within a setpoint range, to optimize energyperformance (e.g., to minimize energy consumption, to minimize energycost, etc.), and/or to satisfy any constraint or combination ofconstraints as may be desirable for various implementations.

Referring now to FIG. 5, a flowchart of a process 500 for determining anuncertainty in parameters of a building energy use model is shown,according to an exemplary embodiment. In some embodiments, process 500is performed by building analysis system 110 using processing circuit304. Process 500 illustrates an empirical process for determining theuncertainties in a vector {circumflex over (θ)} of energy use modelparameters. Vector {circumflex over (θ)} may include estimates for oneor more balance point parameters (i.e., a cooling balance pointparameter {circumflex over (T)}_(bC) and/or a heating balance pointparameter {circumflex over (T)}_(bH)) and one or more coefficients of abuilding energy use model (e.g., regression model coefficients{circumflex over (β)}).

Process 500 estimates the uncertainties in model parameters {circumflexover (θ)} using an empirical estimation technique. The empiricalestimation technique may utilize a re-sampling method (e.g., a Bootstrapmethod, a Jackknife method, etc.) to generate multiple samples from theset of building performance data stored in building data module 310 andpredictor variable module 312. For each of the multiple samples, anestimate of the model parameters {circumflex over (θ)} may bedetermined. The uncertainty in the model parameters {circumflex over(θ)} can then be calculated using the multiple estimates of the modelparameters {circumflex over (θ)}.

Still referring to FIG. 5, process 500 is shown to include receiving anenergy use model for a building site (step 502). The energy use modelmay be of any form including parametric models (e.g., linear regressionmodels, non-linear regression models, etc.), non-parametric models(neural networks, kernel estimation, hierarchical Bayesian, etc.), orsomething in between (e.g., Gaussian process models). In someembodiments, step 502 includes receiving an energy use model from anexternal source (e.g., via communications interface 302). In otherembodiments, step 502 includes generating the energy use model (e.g.,using the building data stored in building data module 310 and/or thepredictor variable data stored in predictor variable module 312).

In some embodiments, linear regression is used in step 502 to generatethe energy use model. The energy use model may include one or morepredictor variables and one or more model parameters. A linearregression model for a building may be represented by the followingequation:

Y=Xβ+e,

where Y is the building's energy consumption, X is a predictor variablematrix, β is a vector of unknown regression coefficients (e.g., β₀β₁, .. . β_(p)), and e is the model error such that e˜N(0,t_(k)σ²).

The predictor variable matrix X may have a size of n by p+1 where n isthe total number of observations and where the number p+1 includes ppredictor variables (e.g., x₁ . . . x_(p)) and a time variable trepresenting the duration of each observation period (e.g., t=[t₁ . . .t_(n)]^(T)). For example, the estimated energy consumption during thek^(th) observation period can be expressed as:

ŷ _(k)=β₀ t _(k)+β₁ x _(1,k)+β₂ x _(2,k)+ . . . +β_(p) x _(p,k) +e _(k),

where t_(k) is the time duration of the k^(th) observation period k=1 .. . n.

In some embodiments, the energy use model includes a weather-relatedpredictor variable (e.g., outside air temperature T_(OA), enthalpy,cooling degree days, heating degree days, heating energy days, coolingenergy days, etc.). In some embodiments, the energy use model includesonly one weather-related predictor variable. In some embodiments, theenergy use model includes one or more non-weather-related predictorvariables. Non-weather-related predictor variables may include, forexample, water consumption, building occupancy, days off, the number ofdays per period t, and/or any other variable which may affect the site'senergy consumption. The weather-related predictor variable and/or otherpredictor variables may be included in predictor variable matrix X.

Still referring to FIG. 5, process 500 is shown to include obtaining aset of data points, each of the data points including a value for theone or more predictor variables and an associated energy consumptionvalue for the building site (step 504). The predictor variables mayinclude one or more independent variables of the energy use model (e.g.,predictor variables x_(1,k) . . . x_(p,k), time period duration t_(k),etc.). The energy consumption value for the building site may be theenergy consumption value y_(k). In some embodiments, the values for thepredictor variables may be retrieved from predictor variable module 312and the energy consumption values may be retrieved from building datamodule 310. In some embodiments, the set of data points includes n datapoints, each data point corresponding to an observation of theindependent variable y_(k) and the dependent variables x_(1,k) . . .x_(p,k) and t_(k), where k=1 . . . n.

In some embodiments, the set of data points may be represented by apredictor variable matrix X and an energy consumption vector Y. Thepredictor variable matrix X may be a n by p+1 matrix includingcalculated or measured values for each of p predictor variables (i.e.,x_(1,k) . . . x_(p,k)) and a time variable t_(k). The value of timevariable t_(k) may identify a time duration associated with eachobservation period k. The predictor variable values and/or predictorvariable matrix may be stored in predictor variable module 312. Theenergy consumption vector Y may include an energy consumption valuey_(k) associated with each observation period k, where k=1 . . . n. Eachrow of the predictor variable matrix X and energy consumption vector Ymay correspond to a different data point or observation of the dependentand independent variables.

Still referring to FIG. 5, process 500 is shown to include generatingmultiple samples from the set of data points (step 506). Each of thesamples may include a plurality of data points selected from the set ofdata points. In some embodiments, step 506 includes using a Bootstrapsampling process to generate the multiple samples. Using the Bootstrapsampling process, step 506 may include replacing a predetermined numberof data points in the set of data points with an equal number of datapoints selected randomly from the original set of data points (i.e.,sampling with replacement). The Bootstrap sampling process may produce asample having the same number of data points as the original set (e.g.,n data points), with some of the data points repeated multiple times.Step 506 may include performing the Bootstrap sampling process multipletimes to generate multiple samples from the set of data points.

In some embodiments, step 506 includes using a Jackknife samplingprocess to generate the multiple samples. Using the Jackknife samplingprocess, step 506 may include removing one of the data points from theoriginal set of n data points to generate a sample of n−1 data points.Step 506 may include performing the Jackknife sampling process n times,removing a different data point each time. Using the Jackknife samplingprocess, performing step 506 may generate n samples of n−1 data points.The i^(th) Jackknife sample may include all of the data points in theoriginal set of data points, except for the i^(th) data point, where i=1. . . n. Both the Bootstrap and the Jackknife method may be similar tothe Monte Carlo method. However, with the Bootstrap and Jackknifemethods, there is no assumption of any particular distribution.

Still referring to FIG. 5, process 500 is shown to include, for each ofthe multiple samples, estimating the model parameters {circumflex over(θ)} using the plurality of data points associated with the sample (step508). In some embodiments, step 508 is performed by empirical analysismodule 326, as described with reference to FIG. 3. In some embodiments,step 508 includes estimating the model parameters {circumflex over (θ)}in multiple stages. For example, step 508 is shown to include estimatingthe balance point or points for a particular data sample (step 510),calculating the values for any predictor variables that depend on abalance point (step 512), and determining the regression modelcoefficients {circumflex over (β)}(step 514).

In some embodiments, step 510 includes using outside air temperature andenergy consumption data to estimate the balance points (e.g.,{circumflex over (T)}_(bC) and/or {circumflex over (T)}_(bH)). Forexample, the energy consumption and outside air temperature data may beanalyzed to identify a maximum outside air temperature (i.e., forheating balance point {circumflex over (T)}_(bH)) or a minimum outsideair temperature (i.e., for cooling balance point {circumflex over(T)}_(bC)) at which building energy consumption is a function of theoutside air temperature.

According to another embodiment, step 510 includes using an optimizationscheme to determine the balance point or points. The optimization schememay include an exhaustive search of the balance points, a gradientdescent algorithm, and/or a generalized reduced gradient (GRG) method toestimate balance points {circumflex over (T)}_(bC) and/or {circumflexover (T)}_(bH). In some embodiments, step 510 includes identifying thebalance points using an iteratively reweighted least squares regressionmethod. For example, step 510 may include searching for balance pointsthat minimize the sum of squared error RSS of the building energy usemodel, where

${RSS} = {{\sum\limits_{i = 1}^{n}\; {\frac{{\hat{e}}_{i}^{2}}{t_{i}}\mspace{14mu} {and}\mspace{14mu} \hat{e}}} = {{{Y - {X\; \hat{\beta}}}}.}}$

Step 510 may be performed multiple times to estimate balance pointvalues for each of the multiple samples generated in step 506.

In some embodiments, step 508 includes using the estimated balancepoints {circumflex over (T)}_(bC) and/or {circumflex over (T)}_(bH) tocalculate values for any predictor variables that depend on a balancepoint (step 512). Such predictor variables may be, for example, coolingdegree days (CDD) and/or heating degree days (HDD), if the balancepoints are temperature balance points or cooling energy days (CED)and/or heating energy days (HED) if the balance points are enthalpybalance points. For embodiments in which predictor variable x_(1,k)corresponds to cooling degree days and predictor variable x_(2,k)corresponds to heating degree days, step 512 may include may calculatingvalues for predictor variables x_(1,k) and x_(2,k) using the followingequations:

x_(1, k) = f(T̂_(bC)) = ∫_(t_(k))max (0, T_(OA)(t) − T̂_(bC)) tx_(2, k) = f(T̂_(bH)) = ∫_(t_(k))max (0, T̂_(bH) − T_(OA)(t)) t.

In some embodiments, step 508 includes determining values for theregression model coefficients {circumflex over (β)}(step 514). Step 514may include using the predictor variable values x_(1,k) . . . x_(p,k)and the associated energy consumption values y_(k) to calculate theregression model coefficients) {circumflex over (β)}. In step 514, anyof a variety of different estimation techniques may be used to estimateregression model coefficients {circumflex over (β)}. In someembodiments, step 514 includes using a partial least squares regression(PLSR) method. In other embodiments, step 514 may include using othermethods, such as ridge regression (RR), principal component regression(PCR), weighted least squares regression (WLSR), or ordinary leastsquares regression (OLSR). The optimal value of) {circumflex over (β)}based on a least squares estimation has the solution {circumflex over(β)}=(X^(T)X)⁻¹ X^(T)Y.

Still referring to FIG. 5, process 500 is shown to include determiningan uncertainty in the model parameters using the multiple estimates ofthe model parameters (step 516). In some embodiments, step 514 includesusing the estimated balance points for each of the multiple samples tocalculate an uncertainty in the balance point estimates. For example,the uncertainty in a balance point estimate {circumflex over (T)}_(b)(i.e., {circumflex over (T)}_(bC) or {circumflex over (T)}_(bH)) can beobserved by examining the mean and standard deviation of the set ofbalance point estimates obtained from the multiple samples generated bydata sampling module 324. The mean T_(b) and standard deviation σ_(T)_(b) of the set of balance point estimates {circumflex over (T)}_(b,1) .. . {circumflex over (T)}_(b,N) can be determined using the followingequations:

${\overset{\_}{T}}_{b} = {\frac{1}{N}{\sum\limits_{j = 1}^{N}\; {\hat{T}}_{b,j}}}$${\sigma_{T_{b}} = \sqrt{\frac{1}{N}{\sum\limits_{j = 1}^{N}\; ( {{\hat{T}}_{b,j} - {\overset{\_}{T}}_{b}} )^{2}}}},$

where N is the number of samples generated by data sampling module 324and {circumflex over (T)}_(b,j) is the j^(th) balance point estimate,j=1 . . . N.

In some embodiments, step 514 includes calculating the uncertainties ofregression model coefficients {circumflex over (β)} using a similarprocess. In other embodiments, the uncertainties in regression modelcoefficients {circumflex over (β)} are estimated using the analyticaltechnique described with reference to FIG. 4. Advantageously, theBootstrap and Jackknife methods embodied in process 500 may be used todetermine a balance point uncertainty and to provide informationregarding how the balance point or points are distributed.

In some embodiments, process 500 includes performing a multivariateuncertainty analysis of two or more model parameters {circumflex over(θ)} (e.g., a balance point parameter and a regression modelcoefficient). The regression model coefficient included in themultivariate uncertainty analysis may be associated with aweather-related predictor variable that is a function of the balancepoint parameter. The multivariate uncertainty analysis may serve as atool for visualizing a correlation between model parameters (assuming aprobability distribution of the balance point parameter and thecorresponding regression model coefficient). Such a multivariateuncertainty analysis is not possible using traditional regressiontechniques which only calculate the uncertainties in the regressionmodel coefficients {circumflex over (β)}.

Referring now to FIG. 6, a flowchart of a process 600 for analyzing theenergy performance of a building site is shown, according to anexemplary embodiment. In some embodiments, process 600 is performed bybuilding analysis system 110 using processing circuit 304. In someembodiments, process 600 is performed subsequent to process 400 and/orprocess 500 and uses the results thereof. For example, process 600 mayuse the estimated model parameters {circumflex over (θ)} (i.e., balancepoint parameters {circumflex over (T)}_(bC) and/or {circumflex over(T)}_(bH) and estimated regression model coefficients {circumflex over(β)}) and their associated uncertainties to analyze the energyperformance of a building site.

Process 600 is shown to include determining an uncertainty in a breakeven temperature parameter of an energy use model for a building site(step 602). The break even temperature parameter may be a heatingbalance point {circumflex over (T)}_(bH) or a cooling balance point{circumflex over (T)}_(bC). Step 602 may be accomplished by performingprocess 400 and/or process 500 as previously described with reference toFIGS. 4-5.

In some embodiments, step 602 includes simultaneously determining theuncertainty of the break even temperature parameter and an uncertaintyof one or more additional parameters of the energy use model.Advantageously, simultaneously determining the uncertainties of both thebreak even temperature parameters and the one or more additionalparameters may increase the accuracy of the estimated parameter values,thereby improving the accuracy of any subsequent energy performanceanalysis.

Still referring to FIG. 6, process 600 is shown to further include usingthe uncertainty in the break even temperature parameter to analyze theenergy performance of the building site (step 604). Step 604 may be aspecific implementation of step 410 as previously described withreference to FIG. 4. For example, step 604 may include generating energysavings estimates, detecting outlier building sites with poor energyperformance (e.g., by comparing the model parameters and/or modelparameter uncertainties of similar building sites), detecting faults ina building site (e.g., by performing a peer analysis and/or a temporalanalysis of the model parameters), determining the effects of a fault ona building's energy consumption, and/or performing other energy analysisfunctions.

In some embodiments, step 604 includes performing an energy analysisusing a minimal amount of building performance data (e.g., a LEAN energyanalysis). In step 604, the values and associated uncertainties ofparameters in the building energy use model (e.g., the parameters inparameter vector {circumflex over (θ)}) may be used to arrive atconclusions regarding a building's energy performance.

In some embodiments, step 604 includes performing outlier detection(step 606). For example, step 606 may include comparing one or morestatistics of a test building to the probability distribution of thosestatistics for other buildings in the same class (e.g., buildings havingsimilar usage characteristics, buildings located in similar geographicregions, etc.) or to the same building from a different time period. Invarious embodiments, the statistics may be model parameters in parametervector {circumflex over (θ)}, values calculated from the modelparameters {circumflex over (θ)}, energy consumption statistics,normalized energy consumption (e.g., energy consumption per unit area orvolume of the building site), or any other statistic generated from themodel parameters {circumflex over (θ)}, uncertainties in the modelparameters, and/or performance data associated with a particularbuilding site.

In some embodiments, step 606 includes performing a peer analysis of oneor more test statistics for a class of buildings including the buildingsite (step 608). The peer analysis may include calculating a differencebetween an energy use model parameter for the building site and a meanof the energy use model parameters for the class of buildings. Theenergy use model parameter may be a function of the break eventemperature parameter (e.g., CDD, HDD, etc.), the break even temperatureparameter itself, or another parameter in the building energy use model.In some embodiments, step 608 includes using the uncertainty in thebreak even temperature parameter to improve the accuracy of thecalculation. For example, the uncertainty in the break even temperaturemay be used to more accurately determine the values for the energy usemodel parameter compared in the peer analysis. In some embodiments, step608 includes detecting an outlier model parameter or an outlier buildingsite based on the result of the peer analysis.

In some embodiments, step 606 includes performing a temporal analysis ofone or more test statistics for the building site (step 610). Thetemporal analysis may include calculating a difference between a valuefor an energy use model parameter at a particular time and a mean of aset of values for the energy use model parameter. The set of values mayinclude a plurality of values for the energy use model parameter atvarious times for the same building site. In some embodiments, step 610includes detecting an outlier model parameter based on a result of thetemporal analysis.

In some embodiments, step 604 includes detecting the existence of afault condition (step 612). In some embodiments, step 612 includesmonitoring the energy use model parameters for a change and detectingthe fault condition in response to an observed change in one or more ofthe monitored parameters. Advantageously, the systems and methods of thepresent disclosure may improve the accuracy and/or certainty of thefault detection in step 612 by more accurately estimating the values forthe uncertainties in the model parameters {circumflex over (θ)}.

In some embodiments, step 604 includes determining an effect of a faulton a buildings energy consumption, calculating an energy savingsopportunity for the building site, and/or using the model parameters{circumflex over (θ)} to detect excessive energy consumption. Using thesystems and methods described herein, the model parameters {circumflexover (θ)} and their associated uncertainties can be determined moreaccurately than with traditional techniques, thereby resulting in moreaccurate energy savings estimates, outlier detection, and/or faultdetection in process 600.

Embodiments of the subject matter and the operations described in thisspecification can be implemented in digital electronic circuitry, or incomputer software embodied on a tangible medium, firmware, or hardware,including the structures disclosed in this specification and theirstructural equivalents, or in combinations of one or more of them.Embodiments of the subject matter described in this specification can beimplemented as one or more computer programs, i.e., one or more modulesof computer program instructions, encoded on one or more computerstorage medium for execution by, or to control the operation of, dataprocessing apparatus. Alternatively or in addition, the programinstructions can be encoded on an artificially-generated propagatedsignal, e.g., a machine-generated electrical, optical, orelectromagnetic signal, that is generated to encode information fortransmission to suitable receiver apparatus for execution by a dataprocessing apparatus. A computer storage medium can be, or be includedin, a computer-readable storage device, a computer-readable storagesubstrate, a random or serial access memory array or device, or acombination of one or more of them. Moreover, while a computer storagemedium is not a propagated signal, a computer storage medium can be asource or destination of computer program instructions encoded in anartificially-generated propagated signal. The computer storage mediumcan also be, or be included in, one or more separate components or media(e.g., multiple CDs, disks, or other storage devices). Accordingly, thecomputer storage medium may be tangible and non-transitory.

The operations described in this specification can be implemented asoperations performed by a data processing apparatus on data stored onone or more computer-readable storage devices or received from othersources.

The term “client or “server” include all kinds of apparatus, devices,and machines for processing data, including by way of example aprogrammable processor, a computer, a system on a chip, or multipleones, or combinations, of the foregoing. The apparatus can includespecial purpose logic circuitry, e.g., an FPGA (field programmable gatearray) or an ASIC (application-specific integrated circuit). Theapparatus can also include, in addition to hardware, code that createsan execution environment for the computer program in question, e.g.,code that constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, a cross-platform runtimeenvironment, a virtual machine, or a combination of one or more of them.The apparatus and execution environment can realize various differentcomputing model infrastructures, such as web services, distributedcomputing and grid computing infrastructures.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, object, orother unit suitable for use in a computing environment. A computerprogram may, but need not, correspond to a file in a file system. Aprogram can be stored in a portion of a file that holds other programsor data (e.g., one or more scripts stored in a markup languagedocument), in a single file dedicated to the program in question, or inmultiple coordinated files (e.g., files that store one or more modules,sub-programs, or portions of code). A computer program can be deployedto be executed on one computer or on multiple computers that are locatedat one site or distributed across multiple sites and interconnected by acommunication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform actions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read-only memory ora random access memory or both. The essential elements of a computer area processor for performing actions in accordance with instructions andone or more memory devices for storing instructions and data. Generally,a computer will also include, or be operatively coupled to receive datafrom or transfer data to, or both, one or more mass storage devices forstoring data, e.g., magnetic, magneto-optical disks, or optical disks.However, a computer need not have such devices. Moreover, a computer canbe embedded in another device, e.g., a mobile telephone, a personaldigital assistant (PDA), a mobile audio or video player, a game console,a Global Positioning System (GPS) receiver, or a portable storage device(e.g., a universal serial bus (USB) flash drive), to name just a few.Devices suitable for storing computer program instructions and datainclude all forms of non-volatile memory, media and memory devices,including by way of example semiconductor memory devices, e.g., EPROM,EEPROM, and flash memory devices; magnetic disks, e.g., internal harddisks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROMdisks. The processor and the memory can be supplemented by, orincorporated in, special purpose logic circuitry.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube), LCD (liquidcrystal display), OLED (organic light emitting diode), TFT (thin-filmtransistor), plasma, other flexible configuration, or any other monitorfor displaying information to the user and a keyboard, a pointingdevice, e.g., a mouse, trackball, etc., or a touch screen, touch pad,etc., by which the user can provide input to the computer. Other kindsof devices can be used to provide for interaction with a user as well;for example, feedback provided to the user can be any form of sensoryfeedback, e.g., visual feedback, auditory feedback, or tactile feedback;and input from the user can be received in any form, including acoustic,speech, or tactile input. In addition, a computer can interact with auser by sending documents to and receiving documents from a device thatis used by the user; for example, by sending web pages to a web browseron a user's client device in response to requests received from the webbrowser.

Embodiments of the subject matter described in this specification can beimplemented in a computing system that includes a back-end component,e.g., as a data server, or that includes a middleware component, e.g.,an application server, or that includes a front-end component, e.g., aclient computer having a graphical user interface or a Web browserthrough which a user can interact with an embodiment of the subjectmatter described in this specification, or any combination of one ormore such back-end, middleware, or front-end components. The componentsof the system can be interconnected by any form or medium of digitaldata communication, e.g., a communication network. Examples ofcommunication networks include a local area network (“LAN”) and a widearea network (“WAN”), an inter-network (e.g., the Internet), andpeer-to-peer networks (e.g., ad hoc peer-to-peer networks).

While this specification contains many specific embodiment details,these should not be construed as limitations on the scope of anyinventions or of what may be claimed, but rather as descriptions offeatures specific to particular embodiments of particular inventions.Certain features that are described in this specification in the contextof separate embodiments can also be implemented in combination in asingle embodiment. Conversely, various features that are described inthe context of a single embodiment can also be implemented in multipleembodiments separately or in any suitable subcombination. Moreover,although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the embodiments described above should not be understoodas requiring such separation in all embodiments, and it should beunderstood that the described program components and systems cangenerally be integrated together in a single software product embodiedon a tangible medium or packaged into multiple such software products.

Thus, particular embodiments of the subject matter have been described.Other embodiments are within the scope of the following claims. In somecases, the actions recited in the claims can be performed in a differentorder and still achieve desirable results. In addition, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In certain embodiments, multitasking and parallel processingmay be advantageous.

The background section is intended to provide a background or context tothe invention recited in the claims. The description in the backgroundsection may include concepts that could be pursued, but are notnecessarily ones that have been previously conceived or pursued.Therefore, unless otherwise indicated herein, what is described in thebackground section is not prior art to the description or claims and isnot admitted to be prior art by inclusion in the background section.

What is claimed is:
 1. A method for determining an uncertainty inparameters of a building energy use model, the method comprising:receiving an energy use model for a building site, the energy use modelhaving one or more model parameters, wherein the energy use model isused to predict an energy consumption of the building site as a functionof the one or more model parameters and one or more predictor variables;calculating a gradient of an output of the energy use model with respectto the model parameters; determining a covariance matrix using thecalculated gradient; and using the covariance matrix to identify anuncertainty of the model parameters.
 2. The method of claim 1, whereinthe uncertainty of the model parameters is a function of entries in thecovariance matrix.
 3. The method of claim 1, wherein at least one of themodel parameters is a balance point parameter defining a range ofoutside air temperatures within which the output of the energy use modeldepends on the outside air temperature.
 4. The method of claim 1,wherein at least one of the predictor variables is a function of abalance point parameter of the energy use model, wherein determining thecovariance matrix comprises: identifying, in the energy use model, apredictor variable that is a function of the balance point parameter;identifying a regression coefficient associated with the identifiedpredictor variable; and using a function of the identified regressioncoefficient as a gradient of the output of the energy use model withrespect to the balance point parameter.
 5. The method of claim 4,wherein the balance point is a threshold temperature value defining arange of outside air temperatures within which the output of the energyuse model depends on the outside air temperature; wherein the functionof the identified regression coefficient is a product of the identifiedregression coefficient and a variable representing a total time durationduring which the outside air temperature is within the outside airtemperature range.
 6. The method of claim 1, wherein determining thecovariance matrix comprises determining a standard error of regressionfor the energy use model.
 7. The method of claim 1, further comprising:obtaining a plurality of data points, each of the data points comprisinga value for the one or more predictor variables and an associated energyconsumption value for the building site; and estimating the modelparameters using the plurality of data points.
 8. The method of claim 7,wherein obtaining the plurality of data points comprises, for each ofthe data points: receiving at least one of: an observed temperaturevalue and an observed enthalpy value; and calculating the value of thepredictor variable using the observed temperature value or the observedenthalpy value.
 9. The method of claim 1, wherein the one or morepredictor variables comprise at least one weather-related predictorvariable, wherein the weather-related predictor variable is at least oneof: cooling degree days, heating degree days, cooling energy days,heating energy days, temperature, and enthalpy.
 10. The method of claim1, wherein the energy use model is at least one of: a non-linear energyuse model, a piecewise linear energy use model, a three-parameter energyuse model, and a five-parameter energy use model.
 11. The method ofclaim 1, further comprising: updating the uncertainty of the modelparameters; applying inputs to the energy use model; conducting aperformance analysis using the energy use model, wherein a result of theperformance analysis is a function of one or more of the modelparameters; providing an output using the result of the performanceanalysis; and determining an uncertainty of the output using the updateduncertainty of the model parameters.
 12. The method of claim 1, furthercomprising: using the uncertainty of the model parameters to perform amultivariate uncertainty analysis of the model parameters.
 13. Themethod of claim 12, wherein the multivariate uncertainty analysis allowsfor a visualization of a correlation between two or more of the modelparameters.
 14. The method of claim 12, wherein the multivariateuncertainty analysis allows for performing a multivariate peer analysisof the model parameters.
 15. A method for determining an uncertainty inparameters of a building energy use model, the method comprising:receiving an energy use model for a building site, the energy use modelhaving one or more model parameters, wherein the energy use model isused to predict an energy consumption of the building site as a functionof the one or more model parameters and one or more predictor variables;obtaining a set of data points, each of the data points comprising avalue for the one or more predictor variables and an associated energyconsumption value for the building site; generating multiple samplesfrom the set of data points, each of the samples comprising a pluralityof data points selected from the set of data points; for each of themultiple samples, estimating the model parameters using the plurality ofdata points associated with the sample; and determining an uncertaintyin the model parameters using the multiple estimates of the modelparameters.
 16. The method of claim 15, wherein obtaining the set ofdata points comprises, for each of the data points: receiving at leastone of: an observed temperature value and an observed enthalpy value;and calculating a value of one or more of the predictor variables usingthe observed temperature value or the observed enthalpy value.
 17. Themethod of claim 15, wherein the one or more predictor variables compriseat least one weather-related predictor variable, wherein theweather-related predictor variable is at least one of: cooling degreedays, heating degree days, cooling energy days, heating energy days,temperature, and enthalpy.
 18. The method of claim 15, wherein at leastone of the model parameters is a balance point parameter defining arange of outside air temperatures within which an output of the energyuse model depends on the outside air temperature.
 19. The method ofclaim 15, wherein generating multiple samples from the set of datapoints comprises, for each of the multiple samples: replacing apredetermined number of data points in the set of data points with anequal number of data points selected randomly from the set of datapoints; wherein each of the multiple samples includes a same number ofdata points as the set of multiple data points.
 20. The method of claim15, wherein generating multiple samples from the set of data pointscomprises, for each of the multiple samples: removing a predeterminednumber of data points from the set of multiple data points; wherein eachof the multiple samples includes less data points than the set of datapoints.
 21. The method of claim 15, further comprising: updating theuncertainty of the model parameters; applying inputs to the energy usemodel; conducting a performance analysis using the energy use model,wherein a result of the performance analysis is a function of one ormore of the model parameters; providing an output using a result of theperformance analysis; and determining an uncertainty of the output usingthe updated uncertainty of the model parameters.
 22. A method foranalyzing an energy performance of a building site, the methodcomprising: determining an uncertainty in a break even temperatureparameter of an energy use model for the building site; and using theuncertainty in the break even temperature parameter to analyze theenergy performance of the building site.
 23. The method of claim 22,wherein determining the uncertainty in the break even temperatureparameter comprises: simultaneously determining the uncertainty of thebreak even temperature parameter and an uncertainty of one or moreadditional parameters of the energy use model; and using thesimultaneously determined uncertainties to improve an accuracy of theenergy performance analysis.
 24. The method of claim 22, whereinanalyzing the energy performance of the building site comprisesperforming a peer analysis of one or more energy use model parametersfor a class of buildings including the building site, the peer analysiscomprising: calculating a difference between an energy use modelparameter for the building site and a mean of the energy use modelparameters for the class of buildings; using the uncertainty in thebreak even temperature parameter to improve an accuracy of thecalculation; and detecting an outlier model parameter based on a resultof the calculation.
 25. The method of claim 22, wherein analyzing theenergy performance of the building site comprises performing a temporalanalysis of one or more energy use model parameters for the buildingsite, the temporal analysis comprising: calculating a difference betweena value for an energy use model parameter at a particular time and amean of a set of values for the energy use model parameter, the set ofvalues including a plurality of values for the energy use modelparameter at various times; using the uncertainty in the break eventemperature parameter to improve an accuracy of the calculation; anddetecting an outlier model parameter based on a result of thecalculation.
 26. The method of claim 22, wherein analyzing the energyperformance of the building site comprises: monitoring changes to one ormore parameters in the energy use model for the building site; detectingthe existence of a fault condition using a monitored change to theenergy use model parameters; and using the uncertainty in the break eventemperature parameter to improve an accuracy of the detection.
 27. Themethod of claim 22, wherein analyzing the energy performance of thebuilding site comprises: calculating an energy savings opportunity forthe building site; and using the uncertainty in the break eventemperature parameter to improve an accuracy of the calculation.
 28. Themethod of claim 22, further comprising: updating the uncertainty of themodel parameters; applying inputs to the energy use model; conducting aperformance analysis using the energy use model, wherein a result of theperformance analysis is a function of one or more of the modelparameters; providing an output using the result of the performanceanalysis; and determining an uncertainty of the output using the updateduncertainty of the model parameters.